Fundamentals of Group Theory: An Advanced Approach

Steven Roman

Fundamentals of Group Theory An Advanced Approach

Steven Roman Irvine, CA USA

ISBN 978-0-8176-8300-9 e-ISBN 978-0-8176-8301-6 DOI 10.1007/978-0-8176-8301-6 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011941115 Mathematics Subject Classification (2010): 20-01 © Springer Science+Business Media, LLC 2012 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhäuser Boston, c/o Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper

Birkhäuser Boston is part of Springer Science+Business Media (www.birkhauser.com)

To Donna

Preface

This book is intended to be an advanced look at the basic theory of groups, suitable for a graduate class in group theory, part of a graduate class in abstract algebra or for independent study. It can also be read by advanced undergraduates. Indeed, I assume no specific background in group theory, but do assume some level of mathematical sophistication on the part of the reader. A look at the table of contents will reveal that the overall topic selection is more or less standard for a book on this subject. Let me at least mention a few of the perhaps less standard topics covered in the book: 1) An historical look at how Galois viewed groups. 2) The problem of whether the commutator subgroup of a group is the same as the set of commutators of the group, including an example of when this is not the case. 3) A discussion of xY-groups, in particular, a) groups in which all subgroups have a complement b) groups in which all normal subgroups have a complement c) groups in which all subgroups are direct summands d) groups in which all normal subgroups are direct summands. 4) The subnormal join property, that is, the property that the join of two subnormal subgroups is subnormal. 5) Cancellation in direct sums: A group K is cancellable in direct sums if E { K ¸ F { Lß

K¸L

Ê

E¸F

(The symbol { represents the external direct sum.) We include a proof that any finite group is cancellable in direct sums. 6) A complete proof of the theorem of Baer that a nonabelian group K has the property that all of its subgroups are normal if and only if KœUEF where U is a quaternion group, E is an elementary abelian group of exponent # and F is an abelian group all of whose elements have odd order.

vii

viii

Preface

7) A somewhat more in-depth discussion of the structure of :-groups, including the nature of conjugates in a :-group, a proof that a :-group with a unique subgroup of any order must be either cyclic (for :  #) or else cyclic or generalized quaternion (for : œ #) and the nature of groups of order :8 that have elements of order : 8" . 8) A discussion of the Sylow subgroups of the symmetric group (in terms of wreath products). 9) An introduction to the techniques used to characterize finite simple groups. 10) Birkhoff's theorem on equational classes and relative freeness. Here are a few other remarks concerning the nature of this book. 1) I have tried to emphasize universality when discussing the isomorphism theorems, quotient groups and free groups. 2) I have introduced certain concepts, such as subnormality and chain conditions perhaps a bit earlier than in some other texts at this level, in the hopes that the reader would acclimate to these concepts earlier. 3) I have also introduced group actions early in the text (Chapter 4), before giving a more thorough discussion in Chapter 7. 4) I have emphasized the role of applying certain operations, namely intersection, lifting, quotient and unquotient to a “group extension” L Ÿ K . A couple of random notes: Unless otherwise indicated, any theorem not proved in the text is an invitation to the reader to supply a proof. Also, sections marked with an asterisk are optional, meaning that they can be skipped without missing information that will be required later. Let me conclude by thanking my graduate students of the past five years, who not only put up with this material in manuscript form but also put up with the many last-minute changes that I made to the manuscript during those years. In any case, if the reader should find any errors, I would appreciate a heads-up. I can be contacted through my web site www.romanpress.com. Steven Roman

Contents

Preface, vii

1

Preliminaries, 1 Multisets, 1 Words, 1 Partially Ordered Sets, 2 Chain Conditions and Finiteness, 5 Lattices, 8 Equivalence Relations, 10 Cardinality, 12 Miscellanea, 15

2

Groups and Subgroups, 19 Operations on Sets, 19 Groups, 19 The Order of a Product, 26 Orders and Exponents, 28 Conjugation, 29 The Set Product, 31 Subgroups, 32 Finitely-Generated Groups, 35 The Lattice of Subgroups of a Group, 37 Cosets and Lagrange's Theorem, 41 Euler's Formula, 43 Cyclic Groups, 44 Homomorphisms of Groups, 46 More Groups, 46 *An Historical Perspective: Galois-Style Groups, 56 Exercises, 58

3

Cosets, Index and Normal Subgroups, 61 Cosets and Index, 61 Quotient Groups and Normal Subgroups, 65 Internal Direct Products, 72 Chain Conditions and Subnormality, 76 ix

x

Contents Subgroups of Index #, 78 Cauchy's Theorem, 79 The Center of a Group; Centralizers, 81 The Normalizer of a Subgroup, 82 Simple Groups, 84 Commutators of Elements, 84 Commutators of Subgroups, 93 *Multivariable Commutators, 95 Exercises, 98

4

Homomorphisms, Chain Conditions and Subnormality, 105 Homomorphisms, 105 Kernels and the Natural Projection, 108 Groups of Small Order, 108 A Universal Property and the Isomorphism Theorems, 110 The Correspondence Theorem, 113 Group Extensions, 115 Inner Automorphisms, 119 Characteristic Subgroups, 120 Elementary Abelian Groups, 121 Multiplication as a Permutation, 123 The Frattini Subgroup of a Group, 127 Subnormal Subgroups, 128 Chain Conditions, 136 Automorphisms of Cyclic Groups, 141 Exercises, 144

5

Direct and Semidirect Products, 149 Complements and Essentially Disjoint Products, 149 Product Decompositions, 151 Direct Sums and Direct Products, 152 Cancellation in Direct Sums, 157 The Classification of Finite Abelian Groups, 158 Properties of Direct Summands, 161 xY-Groups, 164 Behavior Under Direct Sum, 167 When All Subgroups Are Normal, 169 Semidirect Products, 171 The External Semidirect Product, 175 *The Wreath Product, 180 Exercises, 185

6

Permutation Groups, 191 The Definition and Cycle Representation, 191 A Fundamental Formula Involving Conjugation, 193 Parity, 194 Generating Sets for W8 and E8 , 195

Contents

xi

Subgroups of W8 and E8 , 197 The Alternating Group Is Simple, 197 Some Counting, 200 Exercises, 202

7

Group Actions; The Structure of : -Groups, 207 Group Actions, 207 Congruence Relations on a K-Set, 211 Translation by K, 212 Conjugation by K on the Conjugates of a Subgroup, 214 Conjugation by K on a Normal Subgroup, 214 The Structure of Finite :-Groups, 215 Exercises, 229

8

Sylow Theory, 235 Sylow Subgroups, 235 The Normalizer of a Sylow Subgroup, 235 The Sylow Theorems, 236 Sylow Subgroups of Subgroups, 238 Some Consequences of the Sylow Theorems, 239 When All Sylow Subgroups Are Normal, 240 When a Subgroup Acts Transitively; The Frattini Argument, 243 The Search for Simplicity, 244 Groups of Small Order, 250 On the Existence of Complements: The Schur–Zassenhaus Theorem, 252 *Sylow Subgroups of W8, 258 Exercises, 261

9

The Classification Problem for Groups, 263 The Classification Problem for Groups, 263 The Classification Problem for Finite Simple Groups, 263 Exercises, 271

10

Finiteness Conditions, 273 Groups with Operators, 273 H-Series and H-Subnormality, 276 Composition Series, 278 The Remak Decomposition, 282 Exercises, 288

11

Solvable and Nilpotent Groups, 291 Classes of Groups, 291 Operations on Series, 293 Closure Properties of Groups Defined by Series, 295 Nilpotent Groups, 297 Solvability, 305 Exercises, 313

xii

12

Contents

Free Groups and Presentations, 319 Free Groups, 319 Relatively Free Groups, 326 Presentations of a Group, 336 Exercises, 350

13

Abelian Groups, 353 An Abelian Group as a ™-Module, 355 The Classification of Finitely-Generated Abelian Groups, 355 Projectivity and the Right-Inverse Property, 357 Injectivity and the Left-Inverse Property, 359 Exercises, 363

References, 367 List of Symbols, 371 Index, 373

Chapter 1

Preliminaries

In this chapter, we gather together some basic facts that will be useful in the text. Much of this material may already be familiar to the reader, so a light skim to set the notation may be all that is required. The chapter then can be used as a reference.

Multisets The following simple concept is much more useful than its infrequent appearance would indicate. Definition Let W be a nonempty set. A multiset Q with underlying set W is a set of ordered pairs Q œ ÖÐ=3 ß 83 Ñ ± =3 − Wß 83 − ™ ß =3 Á =4 for 3 Á 4× where ™ œ Ö"ß #ß á ×. The positive integer 83 is referred to as the multiplicity of the element =3 in Q . A multiset is finite if the underlying set is finite. The size of a finite multiset Q is the sum of the multiplicities of its elements. … For example, Q œ ÖÐ+ß #Ñß Ð,ß $Ñß Ð-ß "Ñ× is a multiset with underlying set W œ Ö+ß ,ß -×. The element + has multiplicity #. One often writes out the elements of a multiset according to their multiplicities, as in Q œ Ö+ß +ß ,ß ,ß ,ß -× Two multisets are equal if their underlying sets are equal and if the multiplicities of each element in the multisets are equal.

Words We will have considerable use for the following concept. Definition Let \ be a nonempty set. A finite sequence = œ ÐB" ß á ß B8 Ñ of elements of \ is called a word or string over \ and is usually written in the form S. Roman, Fundamentals of Group Theory: An Advanced Approach, DOI 10.1007/978-0-8176-8301-6_1, © Springer Science+Business Media, LLC 2012

1

2

Fundamentals of Group Theory

= œ B" âB8 The number of elements in A is the length of A, denoted by lenÐAÑ. There is a unique word of length !, called the empty word and denoted by %. The set of all words over \ is denoted by \ ‡ and \ is called the alphabet for \ ‡ . A subword or substring of a word = is a subsequence of = consisting of consecutive elements of =. The empty word is considered a subword of all words.… The set \ ‡ of words over \ has an algebraic structure. In particular, the operation of juxtaposition (also called concatenation) is associative and has identity %. Any nonempty set with an associative operation that has an identity is called a monoid. Thus, \ ‡ is a monoid under juxtaposition. It is customary to allow the use of exponents other than " when writing words, where B8 œ î BâB

8 factors

for 8  !. Note, however, that this is merely a shorthand notation. Also, it does not affect the length of a word; for example, the length of B# C$ D is '.

Partially Ordered Sets We will need some basic facts about partially ordered sets. Definition A partially ordered set is a pair ÐT ß Ÿ Ñ where T is a nonempty set and Ÿ is a binary relation called a partial order, read “less than or equal to,” with the following properties: 1) (Reflexivity) For all + − T , +Ÿ+ 2) (Antisymmetry) For all +ß , − T , + Ÿ ,ß

,Ÿ+

Ê

+œ,

Ê

+Ÿ-

3) (Transitivity) For all +ß ,ß - − T , + Ÿ ,ß

,Ÿ-

Partially ordered sets are also called posets. … Sometimes partially ordered sets are more easily defined using strict order relations. Definition A strict order  on a nonempty set T is a binary relation that satisfies the following properties:

Preliminaries

3

1) (Asymmetry) For all +ß , − T , +,

Ê

, y +

,-

Ê

2) (Transitivity) For all +ß ,ß - − T , +  ,ß

+-

…

Theorem 1.1 If ÐT ß Ÿ Ñ is a partially ordered set, then the relation +,

if + Ÿ ,ß + Á ,

is a strict order on T . Conversely, if  is a strict order on T , then the relation +Ÿ,

if +  , or + œ ,

is a partial order on T .… It is customary to use a phrase such as “Let T be a partially ordered set” when the partial order is understood. Also, it is very convenient to extend the notation a bit and define W Ÿ + for any subset W of T to mean that = Ÿ + for all = − W . Similarly, + Ÿ W means that + Ÿ = for all = − W and W Ÿ X means that = Ÿ > for all = − W and > − X . Note that in a partially ordered set, it is possible that not all elements are comparable. In other words, it is possible to have Bß C − T with the property that B Ÿ ± C and C Ÿ ± B. Here are some special kinds of partially ordered sets. Definition Let ÐT ß Ÿ Ñ be a partially ordered set. 1) The order Ÿ is called a total order or linear order if every two elements of T are comparable. In this case, ÐT ß Ÿ Ñ is called a totally ordered set or linearly ordered set. 2) A nonempty subset of T that is totally ordered is called a chain in T . The family of chains of T is ordered by set inclusion. 3) A nonempty subset of T for which no two elements are comparable is called an antichain in T . 4) A nonempty subset H of a partially ordered set T is directed if every two elements of H have an upper bound in H.… Definition Let ÐT ß Ÿ Ñ be a poset and let +ß , − T . 1) The closed interval Ò+ß ,Ó is defined by Ò+ß ,Ó œ Ö: − T ± + Ÿ : Ÿ ,× 2) The open interval Ð+ß ,Ñ is defined by Ð+ß ,Ñ œ Ö: − T ± +  :  ,×

4

Fundamentals of Group Theory

3) The half open intervals are defined by Ò+ß ,Ñ œ Ö: − T ± + Ÿ :  ,× …

and

Ð+ß ,Ó œ Ö: − T ± +  : Ÿ ,×

Here are some key terms related to partially ordered sets. Definition (Covering) Let ÐT ß Ÿ Ñ be a partially ordered set. If +ß , − T , then , covers +, written + ¡ , , if + Ÿ , and if there are no elements of T between + and , , that is, if +ŸBŸ,

B œ + or B œ ,

Ê

…

Definition (Maximum and minimum elements) Let ÐT ß Ÿ Ñ be a partially ordered set. 1) A maximal element is an element 7 − T with the property that there is no larger element in T , that is : − Tß7 Ÿ :

Ê

7œ:

A maximum (largest or top) element 7 − T is an element for which T Ÿ7 2) A minimal element is an element 8 − T with the property that there is no smaller element in T , that is : − Tß: Ÿ 8

Ê

:œ8

A minimum (smallest or bottom) element 8 in T is an element for which 8ŸT

…

Definition (Upper and lower bounds) Let ÐT ß Ÿ Ñ be a partially ordered set. Let W be a subset of T . 1) An element ? − T is an upper bound for W if WŸ? The smallest upper bound ? for W , if it exists, is called the least upper bound or join of W and is denoted by lubÐWÑ or 1W . Thus, ? has the property that W Ÿ ? and if W Ÿ B then ? Ÿ B. The join of a finite set W œ Ö+" ß á ß =8 × is also denoted by lubÖ+" ß á ß +8 × or +" ” â ” +8 . 2) An element j − T is a lower bound for W if jŸW The largest lower bound j for W , if it exists, is called the greatest lower bound or meet of W and is denoted by glbÐWÑ or 3W . Thus, j has the property that j Ÿ W and if B Ÿ W then B Ÿ j. The meet of a finite set W œ Ö+" ß á ß +8 × is also denoted by glbÖ+" ß á ß +8 × or +" • â • +8 .…

Preliminaries

5

Note that the join of the empty set g is, by definition, the least upper bound of the elements of g. But every element of T is an upper bound for the elements of g and so the least upper bound is the minimum element of T , if it exists. Otherwise g has no join. Similarly, the meet of the empty set is the greatest lower bound of g and since all elements of T are lower bounds for g, the meet of g is the maximum element of T , if it exists. Now we can state Zorn's lemma, which gives a condition under which a partially ordered set has a maximal element. Theorem 1.2 (Zorn's lemma) If T is a partially ordered set in which every chain has an upper bound, then T has a maximal element.… Zorn's lemma is equivalent to the axiom of choice. As such, it is not subject to proof from the axioms of ZF set theory. Also, Zorn's lemma is equivalent to the well-ordering principle. A well ordering on a nonempty set \ is a total order on \ with the property that every nonempty subset of \ has a least element. Theorem 1.3 (Well-ordering principle) Every nonempty set has a well ordering.…

Order-Preserving and Order-Reversing Maps A function 0 À T Ä U between partially ordered sets is order preserving (also called monotone or isotone) if BŸC

Ê

0B Ÿ 0C

BŸC

Í

0B Ÿ 0C

and an order embedding if

Note that an order embedding is injective, since 0 B œ 0 C implies both 0 B Ÿ 0 C and 0 C Ÿ 0 B, which implies that B Ÿ C and C Ÿ B, that is, B œ C . A surjective order-embedding is called an order isomorphism. Similarly, a function 0 À T Ä U is order reversing (also called antitone) if BŸC

Ê

0B   0C

Í

0B   0C

and an order anti-embedding if BŸC

An order anti-embedding is injective and if it is surjective, then it is called an order anti-isomorphism.

Chain Conditions and Finiteness The chain conditions are a form of finiteness condition on a poset.

6

Fundamentals of Group Theory

Definition Let T be a poset. 1) T has the ascending chain condition (ACC) if it has no infinite strictly ascending sequences, that is, for any ascending sequence :" Ÿ :# Ÿ :$ Ÿ â there is an index 8 such that :85 œ :8 for all 5   !. 2) T has the descending chain condition (DCC) if it has no infinite strictly descending sequences, that is, for any descending sequence :"   :#   :$   â there is an index 8 such that :85 œ :8 for all 5   !. 3) T has both chain conditions (BCC) if T has the ACC and the DCC.… The following characterizations of ACC and DCC are very useful. Definition Let T be a poset. 1) T has the maximal condition if every nonempty subset of T has a maximal element. 2) T has the minimal condition if every nonempty subset of T has a minimal element.… Theorem 1.4 Let T be a poset. 1) T has the ACC if and only if it has the maximal condition. 2) T has the DCC if and only if it has the minimal condition. Proof. Suppose T has the ACC and let W © T be nonempty. Let =" − W . If =" is maximal we are done. If not, then we can pick =# − W such that =#  =" . Continuing in this way, we either arrive at a maximal element in W or we get a strictly increasing ascending chain that does not become constant, which contradicts the ACC. Hence, T has the maximal condition. Conversely, if T has the maximal condition then any ascending sequence in T has a maximal element, at which point the sequence becomes constant. The proof of part 2) is similar.… A poset can express “infinitness” by spreading vertically, via an infinite chain or by spreading horizontally, via an infinite antichain. The next theorem shows that these are the only two ways that a poset can express infiniteness. It also says that if a poset has an infinite chain, then it has either an infinite ascending chain or an infinite descending chain. This theorem will prove very useful to us as we explore chain conditions on subgroups of a group. Theorem 1.5 Let T be a poset. 1) The following are equivalent: a) T has no infinite chains. b) T has both chain conditions.

Preliminaries

7

If these conditions hold, then for any +  , in T , there is a maximal finite chain from + to , . 2) The following are equivalent: a) T has no infinite chains and no infinite antichains. b) T is finite. Proof. It is clear that 1a) implies 1b). For the converse, suppose that T has BCC and let V be an infinite chain. The ACC implies that V has a maximal element B" , which must be maximum in V since V is totally ordered. Then V Ï ÖB" × is an infinite chain and we may select its maximum element B#  B" . Continuing in this way gives an infinite strictly descending chain, a contradiction to the DCC. Hence, 1a) and 1b) are equivalent. If 1a) and 1b) hold, then since Ð+ß ,Ó is nonempty, it has a minimal member +" , whence + ¡ +" is a maximal chain from + to +" . If +"  , , then Ð+" ß ,Ó has a minimal member +# and so + ¡ +" ¡ +# is a maximal chain from + to +# . This cannot continue forever and so must produce a maximal finite chain from + to , . For part 2), assume that T has no infinite chains or infinite antichains but that T is infinite. Using the ACC, we will create an infinite descending chain, in contradiction to the DCC. Since T has the maximal condition, it has a maximal element. Let ` œ Ö73 ± 3 − M× be the set of all maximal elements of T . Denote by Æ B the set of all elements of T that are less than or equal to B. (This is read: down B.) Since ` is a nonempty antichain, it must be finite. Moreover, the ACC implies that T œ .Ð Æ 73 Ñ

and so one of the sets, say Æ 73" , must be infinite. The infinite poset T" œ Ð Æ 73" Ñ Ï Ö73" × also has no infinite antichains and no infinite chains. Thus, we may repeat the above process and select an element 73# − T" such that T# œ Ð Æ 73# Ñ Ï Ö73# × is infinite. Note that 73"  73# . Continuing in this way, we get an infinite strictly descending chain.… The presence of a chain condition on a poset T has consequences for meets and joins. Theorem 1.6 1) Let T be a poset in which every nonempty finite subset has a meet. If T has the DCC, then every nonempty subset of T has a meet.

8

Fundamentals of Group Theory

2) Let T be a poset in which every nonempty finite subset has a join. If T has the ACC, then every nonempty subset of T has a join. Proof. For part 1), let W © T be nonempty. The family of all meets of finite subsets of W has a minimal member 7 in T and the minimality of 7 implies that 7 • + œ 7 for all + − T , that is, 7 Ÿ + for all + − T . Hence, 4

+−W

+œ7−T

We leave proof of part 2) to the reader.…

Lattices Many of the partially ordered sets that we will encounter have a bit more structure. Definition 1) A partially ordered set ÐT ß Ÿ Ñ is a lattice if every two elements of T have a meet and a join. 2) A partially ordered set ÐT ß Ÿ Ñ is a complete lattice if every subset of T has a meet and a join.… Thus, a complete lattice has a maximum element (the join of T ) and a minimum element (the meet of T ). Note that if 0 À T Ä U is an order isomorphism of the lattices T and U, then 0 preserves meets and joins, that is, 0 Š4:3 ‹ œ 40 :3

and

0 Š2:3 ‹ œ 20 :3

However, an order embedding need not preserve these operations. We will often encounter partially ordered sets T for which every subset of elements has a meet. In this case, joins also exist and T is a complete lattice. Theorem 1.7 Suppose that ÐT ß Ÿ Ñ is a partially ordered set for which every subset of T has a meet. Then ÐT ß Ÿ Ñ is a complete lattice, where the join of a subset W of T is the meet of all upper bounds for W . Proof. First, note that the meet of the empty set is the maximum element of T and the meet of T is the minimum element of T and so T is bounded. In particular, the join of g exists. Let W be a nonempty subset of T . The family Y of upper bounds for W is nonempty, since it contains the maximum element. We need only show that the meet 7 œ 3Y is the join of W . Since = Ÿ Y for any = − W , that is, any = − W is a lower bound for Y , it follows that = Ÿ 7, that is, 7   W . Moreover, if 8   W then 8 − Y and so 7 Ÿ 8. Hence, 7 is the least upper bound of W .…

Preliminaries

9

The previous theorem is very useful in many algebraic contexts. In particular, suppose that \ is a nonempty set and that Y is a family of subsets of \ that contains both g and \ and is closed under intersection. (Examples are the subspaces of a vector space, the subgroups of a group, the ideals in a ring, the subfields of a field, the sublattices of a latttice and so on.) Then Y is a complete lattice where the join of any subfamily Z of Y is the intersection of all members of Y containing the members of Z . Note that this join need not be the union of Z , since the union may not be a member of Y . However, if the union of Z is a member of Y , then it will be the join of Z . Example 1.8 1) The set ‘ of real numbers, with the usual binary relation Ÿ , is a partially ordered set. It is also a totally ordered set. It has no maximal elements. 2) The set  œ Ö!ß "ß á × of natural numbers, together with the binary relation of divides, is a partially ordered set. It is customary to write 8 ± 7 to indicate that 8 divides 7. The subset W of  consisting of all powers of # is a totally ordered subset of , that is, it is a chain in . The set T œ Ö#ß %ß )ß $ß *ß #(× is a partially ordered set under ± . It has two maximal elements, namely ) and #(. The subset U œ Ö#ß $ß &ß (ß ""× is a partially ordered set in which every element is both maximal and minimal. The partially ordered set  is a complete lattice but the set of all positive integers under division is a lattice that is not complete. 3) Let W be any set and let c ÐWÑ be the power set of W , that is, the set of all subsets of W . Then c ÐWÑ, together with the subset relation © , is a complete lattice.…

Sublattices The subject of sublattices requires a bit of care, since a nonempty subset W of a lattice P inherits the order of P but not necessarily the meets and joins of P. That is, the meet of a subset X of W may be different when X is viewed as a subset of W than when X is viewed as a subset of P. For example, let P œ Ö"ß #ß $ß 'ß "#× under division and let W œ P Ï Ö'×. Then P and W are both lattices under the same partial order. However, in P we have # ” $ œ ' and in W we have # ” $ œ "#. Let us use the term P-meet to refer to the meet in P, and similarly for join. Definition Let P be a lattice and let Q © P be a nonempty subset of P. 1) Q is a sublattice of P if the Q -meet of any finite nonempty subset W © Q exists and is the same as the P-meet of W , and similarly for join, that is, if 4 Wœ4 W Q P

and

2 Wœ2 W Q P

2) If P is a complete lattice, then Q is a complete sublattice of P if theQ meet of any subset W © Q exists and is the same as the P-meet of W , and

10

Fundamentals of Group Theory similarly for join, that is, if

4 Wœ4 W Q P

and

2 Wœ2 W Q P

…

Theorem 1.9 1) A nonempty subset T of a lattice P is a sublattice of P if and only if the Pmeet and the P-join of any finite nonempty subset E © T are in T . 2) A nonempty subset T of a complete lattice P is a complete sublattice of P if and only if the P-meet and the P-join of any subset E © T are in T .…

Equivalence Relations The concept of an equivalence relation plays a major role in mathematics. Definition Let W be a nonempty set. A binary relation µ on W is called an equivalence relation on W if it satisfies the following conditions: 1) (Reflexivity) For all + − W , +µ+ 2) (Symmetry) For all +ß , − W , +µ,

Ê

,µ+

3) (Transitivity) For all +ß ,ß - − W , + µ ,ß , µ -

Ê

+µ-

…

Definition Let µ be an equivalence relation on W . For + − W , the set of all elements equivalent to + is denoted by Ò+Ó œ Ö, − W ± , µ +× and is called the equivalence class of +.… Theorem 1.10 Let µ be an equivalence relation on W . Then 1) , − Ò+Ó Í + − Ò,Ó Í Ò+Ó œ Ò,Ó 2) For any +ß , − W , we have either Ò+Ó œ Ò,Ó or Ò+Ó ∩ Ò,Ó œ g.… Definition A partition of a nonempty set W is a collection c œ ÖE3 ± 3 − M× of nonempty subsets of W , called the blocks of the partition, for which 1) E3 ∩ E4 œ g for all 3 Á 4 2) W œ -3−M E3 A system of distinct representatives, abbreviated SDR, for a partition c is a set consisting of exactly one element from each block of c . In various contexts, a system of distinct representatives is also called a transversal for c or a set of canonical forms for c .… The following theorem sheds considerable light on the concept of an equivalence relation.

Preliminaries

11

Theorem 1.11 1) Let µ be an equivalence relation on a nonempty set W . Then the set of distinct equivalence classes with respect to µ are the blocks of a partition of W . 2) Conversely, if c is a partition of W , the binary relation µ defined by + µ , if + and , lie in the same block of c is an equivalence relation on W , whose equivalence classes are the blocks of c . This establishes a one-to-one correspondence between equivalence relations on W and partitions of W .… The most important problem related to equivalence relations is that of finding an efficient way to determine when two elements are equivalent. Unfortunately, in most cases, the definition does not provide an efficient test for equivalence and so we are led to the following concepts. Definition Let µ be an equivalence relation on a nonempty set W . A function 0 À W Ä X , where X is any set, is called an invariant of the equivalence relation if it is constant on the equivalence classes, that is, if +µ,

Ê

0 Ð+Ñ œ 0 Ð,Ñ

A function 0 À W Ä X is called a complete invariant if it is constant and distinct on the equivalence classes, that is, if +µ,

Í

0 Ð+Ñ œ 0 Ð,Ñ

A collection Ö0" ß á ß 08 × of invariants is called a complete system of invariants if +µ,

Í

03 Ð+Ñ œ 03 Ð,Ñ for all 3 œ "ß á ß 8

…

Definition Let µ be an equivalence relation on a nonempty set W . A subset G © W is said to be a set of canonical forms for the equivalence relation if G is a system of distinct representatives for the partition consisting of the equivalence classes, that is, if for every = − W , there is exactly one - − G such that - µ =.… A set of canonical forms determines equivalence since +ß , − W are equivalent if and only if their corresponding canonical forms are equal. Of course, this will be a practical solution to the problem of equivalence only if there is a practical way to identify the canonical form associated with each element of W . Often, canonical forms provide more of a theoretical tool than a practical one.

12

Fundamentals of Group Theory

Cardinality Two sets W and X have the same cardinality, written kW k œ kX k

if there is a bijective function (a one-to-one correspondence) between the sets. If W is in one-to-one correspondence with a subset of X , we write kW k Ÿ kX k. If W is in one-to-one correspondence with a proper subset of X but not with X itself, then we write kW k  kX k. The second condition is necessary, since, for instance,  is in one-to-one correspondence with a proper subset of ™ and yet  is also in one-to-one correspondence with ™ itself. Hence, kk œ k™k. This is not the place to enter into a detailed discussion of cardinal numbers. The intention here is that the cardinality of a set, whatever that is, represents the “size” of the set. It is actually easier to talk about two sets having the same, or different, cardinality than it is to define explicitly the cardinality of a given set (a cardinal number is a special kind of ordinal number). For us, it is sufficient simply to associate with each set W a special kind of set known as a cardinal number, denoted by kW k or cardÐWÑ, that is intended to measure the size of the set. In the case of finite sets, the cardinality is the integer that equals the number of elements in the set. Definition 1) A set is finite if it can be put in one-to-one correspondence with a set of the form ™8 œ Ö!ß "ß á ß 8  "×, for some nonnegative integer 8. A set that is not finite is infinite. 2) The cardinal number of the set  of natural numbers is i! (read “aleph nought”), where i is the first letter of the Hebrew alphabet. Hence, kk œ k™k œ kk œ i!

3) Any set with cardinality i! is called a countably infinite set and any finite or countably infinite set is called a countable set. An infinite set that is not countable is said to be uncountable. … Theorem 1.12 ¨ –Bernstein Theorem) For any sets W and X , 1) (Schroder

kW k Ÿ kX k and kX k Ÿ kW k Ê kW k œ kX k

2) (Cantor's theorem) If cÐWÑ denotes the power set of W then kW k  kc ÐWÑk

Preliminaries

13

3) If c! ÐWÑ denotes the set of all finite subsets of W and if W is an infinite set, then kW k œ kc! ÐWÑk

…

Cardinal Arithmetic If W and X are sets, the cartesian product W ‚ X is the set of all ordered pairs W ‚ X œ ÖÐ=ß >Ñ ± = − Wß > − X × If two sets \ and ] are disjoint, their union is called a disjoint union and is denoted by \“] More generally, the disjoint union of two arbitrary sets W and X is the set W “ X œ ÖÐ=ß !Ñ ± = − W× ∪ ÖÐ>ß "Ñ ± > − X × This is just a scheme for taking the union of W and X while at the same time assuring that there is no “collapse” due to the fact that the intersection of W and X may not be empty. Definition Let , and - denote cardinal numbers. Let W and X be sets for which kW k œ , and kX k œ -. 1) The sum ,  - is the cardinal number of the disjoint union W “ X . 2) The product ,- is the cardinal number of W ‚ X . 3) The power ,- is the cardinal number of the set of all functions from X to W.… We will not go into the details of why these definitions make sense. (For instance, they seem to depend on the sets W and X , but in fact they do not.) It can be shown, using these definitions, that cardinal addition and multiplication are associative and commutative and that multiplication distributes over addition. Theorem 1.13 Let ,, - and . be cardinal numbers. Then the following properties hold: 1) (Associativity) ,  Ð-  .Ñ œ Ð,  -Ñ  . and ,Ð-.Ñ œ Ð,-Ñ. 2) (Commutativity) ,  - œ -  , and ,- œ -, 3) (Distributivity) ,Ð-  .Ñ œ ,-  ,.

14

Fundamentals of Group Theory

4) (Properties of Exponents) a ) , - . œ , - , . b) Ð,- Ñ. œ ,-. c) Ð,-Ñ. œ ,. -. … On the other hand, the arithmetic of cardinal numbers can seem a bit strange, as the next theorem shows. Theorem 1.14 Let , and - be cardinal numbers, at least one of which is infinite. Then ,  - œ ,- œ maxÖ,ß -×

…

It is not hard to see that there is a one-to-one correspondence between the power set cÐWÑ of a set W and the set of all functions from W to Ö!ß "×. This leads to the following theorem. Theorem 1.15 For any cardinal , 1) If kW k œ , then kc ÐWÑk œ #, 2) ,  #, … We have already observed that kk œ i! . It can be shown that i! is the smallest infinite cardinal, that is, ,  i0

Ê

, is a natural number

It can also be shown that the set ‘ of real numbers is in one-to-one correspondence with the power set c ÐÑ of the natural numbers. Therefore, k‘k œ #i!

The set of all points on the real line is sometimes called the continuum and so #i! is sometimes called the power of the continuum and denoted by - . The previous theorem shows that cardinal addition and multiplication have a kind of “absorption” quality, which makes it hard to produce larger cardinals from smaller ones. The next theorem demonstrates this more dramatically. Theorem 1.16 1) Addition applied a positive countable number of times or multiplication applied a finite number of times to the cardinal number i! yields i! . Specifically, for any nonzero 8 − , we have i! † i! œ i!

and i8! œ i!

Preliminaries

15

2) Addition and multiplication applied a positive countable number of times to the cardinal number #i! yields #i! . Specifically, i! † #i! œ #i!

and

Ð#i! Ñi! œ #i!

…

Using this theorem, we can establish other relationships, such as #i! Ÿ Ði! Ñi! Ÿ Ð#i! Ñi! œ #i! which, by the Schro¨der–Bernstein theorem, implies that Ði! Ñi! œ #i! We mention that the problem of evaluating ,- in general is a very difficult one and would take us far beyond the scope of this book. We conclude with the following reasonable-sounding result, whose proof is omitted. Theorem 1.17 Let ÖE5 ± 5 − O× be a collection of sets with an index set of cardinality kO k œ ,. If kE5 k Ÿ - for all 5 − O , then » . E5 » Ÿ -,

…

5−O

Miscellanea The following section need not be read until it is referenced much later in the book. If : is prime, then we will have occasion to write an integer α satisfying α ´ " mod : in the form α œ "  ,:> where : y± , . However, the case where : œ # and α œ "  #. with . odd is exceptional. In this case, we will need to write α œ "  ,#> , where , is odd and >   #. This will ensure that >   # when : œ #. Accordingly, it will be useful to introduce the following terminology. Definition Let α ´ " mod :. 1) If : œ # and α œ "  #. where . is odd, that is, if α ´ $ mod %, then the :standard form of α is α œ "  ,:> ß

: y± , and >   #

2) In all other cases, the :-standard form of α is α œ "  ,:> ß

: y± ,

…

Theorem 1.18 Let : be a prime and let .   ". Let 9: Ð8Ñ denote the largest exponent / for which :/ divides 8.

16

Fundamentals of Group Theory

1) For " Ÿ 5 Ÿ :. , 9: ”Œ

:. • œ .  9: Ð5Ñ 5

In particular, :.5" º Œ

:. 5

and if :  # and 5   # or if : œ # and 5   $, then :.5# º Œ

:. 5

2) If the :-standard form of α is α œ /  ,:> then for any .   !, .

.

α: œ /:  A:.> where : y± A. Proof. For part 1), write Œ

:. 5

œ

:. Ð:.  "Ñ Ð:.  ?Ñ :.  Ð5  "Ñ â â 5 " ? 5"

where " Ÿ ? Ÿ 5  ". Now, if " Ÿ 3 Ÿ . , then : 3 ± ? if and only if : 3 ± :.  ? and so :8 ¹ Œ

:. 5

Í

8 Ÿ .  9: Ð5Ñ

The rest follows from the fact that :@ ± 5 implies @ Ÿ 5  " and if :  # and 5   # or if : œ # and 5   $, then :@ ± 5 implies @ Ÿ 5  #. For part 2), if the :-standard form for α is α œ /  ,: > , then α

:.

> :.

œ Ð/  ,: Ñ

:.

œ/ /

:. "

:.

,:

.>

 5œ#

Œ

:. :. 5 5 >5 / , : 5

where the terms in the final sum are ! if . œ !. If :  #, then part 1) implies that the 5 th term in the final sum is divisible by : to the power .  5  #  >5 œ .  >  "  Ò"  >Ð5  "Ñ  5Ó   .  >  " If : œ #, then >   # and so the 5 th term in the final sum is divisible by : to the power

Preliminaries

.  5  "  >5 œ .  >  "  Ò>Ð5  "Ñ  5Ó   .  >  " Hence, in both cases, the final sum is divisible by :.>" and so .

.

α: œ / :  / : where : y± Ð/:

.

"

.

"

,  @:Ñ.…

.

,:.>  @:.>" œ /:  :.> Ð/:

.

"

,  @:Ñ

17

Chapter 2

Groups and Subgroups

Operations on Sets We begin with some preliminary definitions before defining our principal object of study. For a nonemtpy set \ , the 8-fold cartesian product is denoted by \ 8 œ ðóóóñóóóò \‚â‚\ 8 factors

Definition Let \ be a nonempty set and let 8 be a natural number. 1) For 8   ", an 8-ary operation on \ is a function 0 À \8 Ä \ 2) A "-ary operation 0 À \ Ä \ is called a unary operation on \ . 3) A #-ary operation 0 À \ ‚ \ Ä \ is called a binary operation on \ . 4) A nullary operation on \ is an element of \ . An 8-ary operation, for any natural number 8, is referred to as a finitary operation.… It is often the case that the result of applying a binary operation is denoted by juxtaposition, writing +, in place of 0 Ð+ß ,Ñ. Definition If 0 À \ 8 Ä \ is an 8-ary operation on \ and if ] is a nonempty subset of \ , then the restriction of 0 to ] 8 is a map 0 l] 8 À ] 8 Ä \ . We say that ] is closed under the operation 0 if 0 l] 8 maps ] 8 into ] . For a nullary operation B − \ , this means that B − ] .…

Groups We are now ready to define our principal object of study. Definition A group is a nonempty set K, called the underlying set of the group, together with a binary operation on K, generally denoted by juxtaposition, with the following properties: S. Roman, Fundamentals of Group Theory: An Advanced Approach, DOI 10.1007/978-0-8176-8301-6_2, © Springer Science+Business Media, LLC 2012

19

20

Fundamentals of Group Theory

1) (Associativity) For all +, , , - − K, Ð+,Ñ- œ +Ð,-Ñ 2) (Identity) There exists an element " − K, called the identity element of the group, for which "+ œ +" œ + for all + − K. 3) (Inverses) For each + − K, there is an element +" − K, called the inverse of +, for which ++" œ +" + œ " Two elements +ß , − K commute if +, œ ,+ A group is abelian, or commutative, if every pair of elements commute. A group is finite if the underlying set K is a finite set; otherwise, it is infinite. The order of a group is the cardinality of the underlying set K , denoted by 9ÐKÑ or kKk.… It is customary to use the phrase “K is a group” where K is the underlying set when the group operation under consideration is understood. We leave it to the reader to show that the identity element in a group K is unique, as is the inverse of each element. Moreover, for +ß , − K , Ð+" Ñ" œ +

and Ð+,Ñ" œ , " +"

In a group K, exponentiation is defined for integral exponents as follows: Ú Ý Ý" +â+ î 8 + œÛ 8 factors Ý Ý 8 " Ü Ð+ Ñ

if 8 œ ! if 8  ! if 8  !

When K is abelian, the group operation is often (but not always) denoted by  and is called addition, the identity is denoted by ! and called the zero element of the group and the inverse of an element + − K is denoted by + and is called the negative of +. In this case, exponents are replaced by multiples: Ú Ý Ý! +â+ 8+ œ Û ðóóñóóò 8 terms Ý Ý Ü Ð8+Ñ

if 8 œ ! if 8  ! if 8  !

Groups and Subgroups

21

The Order of an Element Definition Let K be a group. 1) If + − K, then any integer 8 for which +8 œ " is called an exponent of +. 2) The smallest positive exponent of + − K, if it exists, is called the order of + and is denoted by 9Ð+Ñ. If + has no exponents, then + is said to have infinite order. An element of finite order is said to be periodic or torsion. 3) An element of order # is called an involution.… Theorem 2.1 Let K be a group. If + − K has finite order 9Ð+Ñ, then the exponents of + are precisely the integral multiples of 9Ð+Ñ. Proof. Let 8 œ 9Ð+Ñ. Any integral multiple of 8 is clearly an exponent of +. Conversely, if +7 œ ", then 7 œ ;8  < where ! Ÿ <  8. Hence, " œ +7 œ +;8< œ +;8 +< œ +< and so the minimality of 8 implies that < œ !, whence 7 œ ;8 is an integral multiple of 8.… Involutions arise often in the theory of groups. Proof of the following result is left as an exercise. Theorem 2.2 A group in which every nonidentity element is an involution is abelian.… As we will see in a moment, a group may have elements of finite order and elements of infinite order. Definition A group K is said to be periodic or torsion if every element of K is periodic. A group that has no periodic elements other than the identity is said to be aperiodic or torsion free.… It is not hard to see that every finite group is periodic. On the other hand, as we will see, there are infinite periodic groups, that is, an infinite group need not have any elements of infinite order.

Examples Here are some examples of groups. Example 2.3 The simplest group is the trivial group K œ Ö"×, which contains only the identity element. All other groups are said to be nontrivial.…

22

Fundamentals of Group Theory

Example 2.4 The integers ™ form an abelian group under addition. The identity is !. The rational numbers  form an abelian group under addition and the nonzero rational numbers ‡ form an abelian group under multiplication. A similar statement holds for the real numbers ‘ and the complex numbers ‚ (and indeed for any field J ).… Example 2.5 (Cyclic groups) If + is a formal symbol, we can define a group K to be the set of all integral powers of +: G∞ Ð+Ñ œ Ö+3 ± 3 − ™× where the product is defined by the formal rules of exponents: +3 +4 œ +34 This group is also denoted by G∞ or Ø+Ù and is called the cyclic group generated by +. The identity of Ø+Ù is " œ +! . We can also create a finite group G8 Ð+Ñ of positive order 8 by setting G8 Ð+Ñ œ Ö" œ +! ß +ß +# ß á ß +8" × where the product is defined by addition of exponents, followed by reduction modulo 8: +3 +4 œ +Ð34Ñ mod 8 This defines a group of order 8, called a cyclic group of order 8. The inverse of +5 is +Ð5Ñ mod 8 . The group G8 Ð+Ñ is also denoted by G8 or Ø+Ù. Note that for any integer 5 , the symbol +5 refers to the element of G8 Ð+Ñ obtained by multiplying together 5 copies of +. Hence, +5 œ +5 mod 8 and so for any integers 5 and 4, +5 +4 œ +5 mod 8 +4 mod 8 œ +5 mod 84 mod 8 œ +Ð54Ñ mod 8 œ +54 Thus, we can feel free to represent the elements of G8 Ð+Ñ using all integral powers of + and the rules of exponents will hold, although we must remember that a single element of G8 Ð+Ñ has many representations as powers of +. The groups G8 Ð+Ñ or G∞ Ð+Ñ are called cyclic groups.… Example 2.6 The set ™8 œ Ö!ß á ß 8  "× of integers modulo a positive integer 8 is a cyclic group of order 8 under addition modulo 8, generated by the element ", since 5 − ™8 is the sum of 5 ones. The notation ™Î8™ is preferred by some mathematicians since when : is a

Groups and Subgroups

23

prime, the notation ™: is also used for the : -adic integers. However, we will not use it in this way. If : is a prime, then the set ™‡: œ Ö"ß á ß :  "× of nonzero elements of ™: is an abelian group under multiplication modulo :. Indeed, by definition, the set J ‡ of nonzero elements of any field J is a group under multiplication. It is possible to prove (and we leave it as an exercise) that J ‡ is cyclic if and only if J is a finite field. More generally, the set V ‡ of units of a commutative ring V with identity is a multiplicative group. In the case of the ring ™8 , this group is ™‡8 œ Ö+ − ™8 ± Ð+ß 8Ñ œ "× To see directly that this is a group, note that for each + − ™‡8 , there exists integers B and C such that B+  C8 œ ". Hence, B+ ´ " mod 8, that is, B − ™‡8 is the inverse of + in ™‡8 . The group ™‡8 is abelian and it is possible to prove (although with some work; see Theorem 4.43) that ™‡8 is cyclic if and only if 8 œ #ß %ß :/ or #:/ , where : is an odd prime.… Example 2.7 (Matrix groups) The set `7ß8 ÐJ Ñ of all 8 ‚ 7 matrices over a field J is an abelian group under addition of matrices. The set KPÐ8ß J Ñ of all nonsingular 8 ‚ 8 matrices over J is a nonabelian (for 8  ") group under multiplication, known as the general linear group. The set WPÐ8ß J Ñ of all 8 ‚ 8 matrices over J with determinant equal to " is a group under multiplication, called the special linear group.… Example 2.8 (Functions) Let K be a group and let \ be a nonempty set. The set K\ of all functions from \ to K is a group under product of functions, defined by Ð0 1ÑÐBÑ œ 0 ÐBÑ1ÐBÑ for all B − \ . The identity in K\ is the function that sends all elements of \ to the identity element " − K . This map is often referred to as the zero map. (We cannot call it the identity map!) Also, the set Y of all bijective functions on K is a group under composition.… Example 2.9 The set K œ ÖÐ/"ß /# ß á Ñ ± /3 − ™# × of all infinite binary sequences, with componentwise addition modulo #, is an infinite abelian group that is periodic, since

24

Fundamentals of Group Theory

#Ð/"ß /# ß á Ñ œ Ð!ß !ß á Ñ and so every nonidentity element has order #.…

The External Direct Product of Groups One important method for creating a new group from existing groups is as follows. If K" ß á ß K8 are groups, then the cartesian product T œ K" ‚ â ‚ K 8 is a group under componentwise product defined by Ð+" ß á ß +8 ÑÐ," ß á ß ,8 Ñ œ Ð+" ," ß á ß +8,8Ñ where +3 ß ,3 − K3 . The group T is called the external direct product of the groups K" ß á ß K8 . Although the notation ‚ is often used for the external direct product of groups, we will use the notation K" } â } K 8 to distinguish it from the cartesian product as a set. As an example, the direct product Z œ G# Ð+Ñ } G# Ð,Ñ œ ÖÐ"ß "Ñß Ð+ß "Ñß Ð"ß ,Ñß Ð+ß ,Ñ× of two cyclic groups of order # is called the (Klein) %-group (and was called the Vierergruppe by Felix Klein in 1884). We will generalize the direct product construction to arbitrary (finite or infinite) families of groups in a later chapter.

Symmetric Groups Let \ be a nonempty set. A bijective function from \ to itself is called a permutation of \ . The set W\ of all permutations of \ is a group under composition, with order k\ kx when \ is finite. Also, W\ is nonabelian for k\ k   $. The group W\ is called the symmetric group or permutation group on the set \ . The group of permutations of the set M8 œ Ö"ß á ß 8× is denoted by W8 and has order 8x. We will study permutation groups in detail in a later chapter, but we want to make a few remarks here for use in subsequent examples. (Proofs will be given later.) If +" ß á ß +5 are distinct elements of \ , the expression Ð+" â+5 Ñ denotes the permutation that sends +3 to +3" for 3 œ "ß á ß 5  " and sends the last element +5 to the first element +" . All other elements of \ are held fixed. This permutation is called a 5 -cycle in W\ . For example, in WÖ"ß#ß$ß%× the permutation Ð" $ %Ñ sends " to $, $ to %, % to " and # to itself. A #-cycle Ð+ ,Ñ is

Groups and Subgroups

25

called a transposition, since it simply transposes + and , , leaving all other elements of \ fixed. We can now see why a permutation group with at least three elements +, , and - is nonabelian, since for example Ð+ ,ÑÐ+ -Ñ Á Ð+ -ÑÐ+ ,Ñ (Composition is generally denoted by juxtaposition as above.) The support of a permutation 5 − W\ is the set of elements of \ that are moved by 5, that is, suppÐ5Ñ œ ÖB − \ ± 5B Á B× Two permutations 5ß 7 − W\ are disjoint if their supports are disjoint. In particular, two cycles Ð+" â+5 Ñ and Ð," â,7 Ñ are disjoint if the underlying sets Ö+" ß á ß +5 × and Ö," ß á ß ,7 × are disjoint. It is not hard to see that disjoint permutations commute, that is, if 5 and 7 are disjoint, then 57 œ 75. It is also not hard to see that every permutation 5 is a product (composition) of pairwise disjoint cycles, the product being unique except for the order of factors and the inclusion of "-cycles. In fact, this is a direct result of the fact that the relation B´C

if

55 B œ C for some 5 − ™

is an equivalence relation and thereby induces a partition on M8 . (The reader is invited to write a complete proof at this time or to refer to Theorem 6.1.) This factorization is called the cycle decomposition of 5 . The cycle structure of 5 is the number of cycles of each length in the cycle decomposition of 5 . For example, the permutation 5 œ Ð" # $ÑÐ% & 'ÑÐ( )ÑÐ*Ñ has cycle structure consisting of two cycles of length $, one cycle of length # and one cycle of length ". It is easy to see that for k\ k   #, any cycle in W\ is a product of transpositions, since Ð+" â+8 Ñ œ Ð+" +8 ÑÐ+" +8" ÑâÐ+" +$ ÑÐ+" +# Ñ and Ð+Ñ œ Ð+ ,ÑÐ+ ,Ñ Hence, the cycle decomposition implies that every permutation is a product of (not necessarily disjoint) transpositions. Although such a factorization is far from unique, we will show that the parity of the number of transpositions in the factorization is unique. In other words, if a permutation can be written as a product of an even number of transpositions, then all factorizations into a

26

Fundamentals of Group Theory

product of transpositions have an even number of transpositions. Such a permutation is called an even permutation. For example, since Ð" $ %Ñ œ Ð" %ÑÐ" $Ñ the permutation Ð" $ %Ñ is even. Similarly, a permutation is odd if it can be written as a product of an odd number of transpositions. For example, the equation above for Ð+" â+8 Ñ shows that a cycle of odd length is an even permutation and a cycle of even length is an odd permutation. One of the most remarkable facts about the permutation groups is that every group has a “copy” that sits inside (is isomorphic to a subgroup of) some permutation group. For example, the Klein %-group sits inside W% as follows: Z œ Ö+ ß Ð" #ÑÐ$ %Ñß Ð" $ÑÐ# %Ñß Ð" %ÑÐ# $Ñ× This is the content of Cayley's theorem, which we will discuss later. Thus, if we knew “everything” about permutation groups, we would know “everything” about all groups!

The Order of a Product One must be very careful not to jump to false conclusions about the order of the product of elements in a group. For example, consider the general linear group KPÐ#ß ‚Ñ of all nonsingular # ‚ # matrices over the complex numbers. Let EœŒ

! "

" !

! and F œ Œ "

" "

We leave it to the reader to show that E and F have finite order but that their product EF has infinite order. On the other extreme, we have 9Ð++" Ñ œ " regardless of the value of 9Ð+Ñ. Thus, the order of a product of two nonidentity elements can be as small as " or as large as infinity. On the other hand, the following key theorem relates the order of a power +5 of an element + − K to the order of +. It also tells us something quite specific about the order of the product of commuting elements. Theorem 2.10 Let K be a group and +ß , − K. 1) If 9Ð+Ñ œ 8, then for " Ÿ 5  8, 8 9Ð+5 Ñ œ gcdÐ8ß 5Ñ In particular, Ø+Ù œ Ø+5 Ù

Í

gcdÐ9Ð+Ñß 5Ñ œ "

Groups and Subgroups

27

2) If 9Ð+Ñ œ 8 and . ± 8, then 9Ð+5 Ñ œ .

Í

8 5 œ < , where gcdÐ<ß .Ñ œ " .

3) If + and , commute, then lcmÐ9Ð+Ñß 9Ð,ÑÑ ¹ 9Ð+,Ñ ¹ lcmÐ9Ð+Ñß 9Ð,ÑÑ gcdÐ9Ð+Ñß 9Ð,ÑÑ In particular, gcdÐ9Ð+Ñß 9Ð,ÑÑ œ "

Ê

9Ð+,Ñ œ 9Ð+Ñ9Ð,Ñ

Proof. For part 1), Theorem 2.1 implies that Ð+5 Ñ7 œ " if and only if 8 ± 57. But 8 ± 57

Í

8 57 ¹ gcdÐ8ß 5Ñ gcdÐ8ß 5Ñ

Í

8 ¹7 gcdÐ8ß 5Ñ

and so the smallest positive exponent 7 of +5 is 8ÎgcdÐ8ß 5Ñ. For part 2), according to part 1), the equation 9Ð+5 Ñ œ . is equivalent to 8 œ. gcdÐ8ß 5Ñ which is equivalent to gcdÐ8ß 5Ñ œ 8Î. , or 8 8 gcdÐ. ß 5Ñ œ . . But this holds if and only if 5 œ <Ð8Î.Ñ with gcdÐ.ß <Ñ œ ". For part 3), let 9Ð+Ñ œ α and 9Ð,Ñ œ " . Then Ð+,Ñ" œ +" has order α 9ÐÐ+,Ñ" Ñ œ 9Ð+" Ñ œ gcdÐαß " Ñ But 9ÐÐ+,Ñ" Ñ divides 9Ð+,Ñ and so

α ¹ 9Ð+,Ñ gcdÐαß " Ñ

A symmetric argument shows that " ¹ 9Ð+,Ñ gcdÐαß " Ñ and since these divisors are relatively prime, we have

28

Fundamentals of Group Theory

α" ¹ 9Ð+,Ñ gcdÐαß " Ñ# But α" ÎÐαß " Ñ# œ lcmÐαß " ÑÎgcdÐαß " Ñ.… Corollary 2.11 Let K be a group and let + − K. If 9Ð+Ñ œ ?@, where Ð?ß @Ñ œ ", then + œ +" +# where +" ß +# − Ø+Ù and 9Ð+" Ñ œ ? and 9Ð+# Ñ œ @. Proof. Since Ð?ß @Ñ œ ", there exist integers = and > for which =?  >@ œ ". Hence, + œ +=?>@ œ +=? +>@ Then Ð=?ß @Ñ œ " implies that 9Ð+=? Ñ œ @ and similarly, 9Ð+>@ Ñ œ ?.…

Orders and Exponents Let K be a group. We have defined an exponent of + − K to be any integer 5 for which +5 œ ". Here is the corresponding concept for subsets of a group. Definition If W © K is nonempty, then an integer 5 for which =5 œ " for all = − W is called an exponent of W . If W has an exponent, we say that W has finite exponent.… Note that many authors reserve the term exponent for the smallest such positive integer 5 . As with individual elements, if the subset W has an exponent, then all exponents of W are multiples of the smallest positive exponent of W . Theorem 2.12 Let K be a group. If a nonempty set W of K has finite exponent, then the set of all exponents of W is the set of all integer multiples of the smallest positive exponent of W .… For a finite group K, the smallest exponent minexpÐKÑ is equal to the least common multiple lcmordersÐKÑ of the orders of the elements of K. Thus, if maxorderÐKÑ denotes the maximum order among the elements of K, then maxorderÐKÑ ± minexpÐKÑ and there are simple examples to show that equality may or may not hold. (The reader is invited to find such examples.) However, in a finite abelian group, equality does hold.

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29

Theorem 2.13 1) If K is a finite group, then maxorderÐKÑ ± minexpÐKÑ œ lcmordersÐKÑ 2) If K is a finite abelian group, then all orders divide the maximum order and so maxorderÐKÑ œ minexpÐKÑ œ lcmordersÐKÑ and K is cyclic if and only if minexpÐKÑ œ 9ÐKÑ. Proof. For the proof of part 2), let + − K have maximum order 7. Suppose to the contrary that there is a , − K for which 9Ð,Ñ y± 7. We will find an element of K of order greater than 7, which is a contradiction. Since 9Ð,Ñ y± 7, there is a prime : for which 9Ð+Ñ œ 7 œ : 4 @

and 9Ð,Ñ œ : 3?

where : y± ?, : y± @ and 3  4. Then 4

9Ð+: Ñ œ

7 :4

and

9Ð, ? Ñ œ :3

Since K is abelian and these orders are relatively prime, we have 4

9Ð+: , ? Ñ œ 7:34  7 as promised. The last statement of the theorem follows from the fact that a finite group K is cyclic if and only if it has an element of order 9ÐKÑ.…

Conjugation Let K be a group. If +ß , − K, then the element , + œ +,+" is called the conjugate of , by +. The conjugacy relation is the binary relation on K defined by +´,

if , œ +B for some B − K

and if + ´ , , we say that + and , are conjugate. Conjugacy is an equivalence " relation on K, since an element is conjugate to itself and if + œ ,B then , œ +B and finally, if , œ +B and - œ , C , then - œ , C œ +BC Note that some authors define the conjugate of , by + as +" ,+, so care must be taken when reading other literature. The function #+ À K Ä K defined by #+ B œ B+

30

Fundamentals of Group Theory

is called conjugation by +. Conjugation is very well-behaved: It is a bijection and preserves the group operation, in the sense that #+ ÐBCÑ œ Ð#+ BÑÐ#+ CÑ Thus, in the language of a later chapter, #+ is a group automorphism. The maps #+ are called inner automorphisms and the set of inner automorphisms is denoted by InnÐKÑ. We will have more to say about InnÐKÑ in later chapters. Theorem 2.14 Let K be a group and let +ß ,ß B − K. Then 1) Conjugation is a bijection that preserves the group operation, that is, ÐB+ Ñ" œ ÐB" Ñ+

and

ÐBCÑ+ œ B+ C+

for all Bß C − K. 2) The conjugation map satisfies ÐB, Ñ+ œ B+, for all B − K. 3) Conjugacy is an equivalence relation on K. The equivalence classes under conjugacy are called conjugacy classes.… We can also apply conjugation to subsets of K. If W © K and + − K, we write #+ W œ W + œ Ö=+ ± = − W× The previous rules generalize to conjugation of sets. In fact, for any Wß X © K and +ß , − K, we have ÐW + Ñ, œ W ,+

and W + œ X + Í W œ X

Conjugation in the Symmetric Group In general, it is not always easy to tell when two elements of a group are conjugate. However, in the symmetric group, it is surprisingly easy. Theorem 2.15 Let W8 be the symmetric group. 1) Let 5 − W8 . For any 5 -cycle Ð+" â+5 Ñ, we have Ð+" â+5 Ñ5 œ Ð5+" â5+5 Ñ Hence, if 7 œ -" â-5 is a cycle decomposition of 7 , then 7 5 œ -"5 â-55 is a cycle decomposition of 7 5 . 2) Two permutations are conjugate if and only if they have the same cycle structure.

Groups and Subgroups

31

Proof. For part 1), we have Ð+" â+5 Ñ5 Ð5+3 Ñ œ œ

5+3" 5+"

35 3œ5

Also, if , Á 5+3 for any 3, then 5" , Á +3 and so Ð+" â+5 Ñ5 , œ 5Ð+" â+5 ÑÐ5" ,Ñ œ 5Ð5" ,Ñ œ , Hence, Ð+" â+5 Ñ5 is the cycle Ð5+" â5+5 Ñ. For part 2), if 7 œ -" â-7 is a cycle decomposition of 7 , then 5 7 5 œ -"5 â-7

and since -35 is a cycle of the same length as -3 , the cycle structure of 7 5 is the same as that of 7 . For the converse, suppose that 5 and 7 have the same cycle structure. If 5 and 7 are cycles, say 5 œ Ð+" â +8 Ñ

and

7 œ Ð," â ,8 Ñ

then any permutation - that sends +3 to ,3 satisfies 5 - œ 7 . More generally, if 5 œ -" â-7

and

7 œ ." â.7

are the cycle decompositions of 5 and 7 , ordered so that -5 has the same length as .5 for all 5 , we can define a permutation - that sends the element in the 3th position of -5 to the element in the 3th position of .5 . Then 5- œ 7 .…

The Set Product It is convenient to extend the group operation on a group K from elements of K to subsets of K. In particular, if W and X are subsets of K, then the set product WX (also called the complex product, since subsets of a group are called complexes in some contexts) is defined by WX œ Ö=> ± = − Wß > − X × As a special case, we write Ö+×W as +W , that is, +W œ Ö+= ± = − W× The set product is associative and distributes over union, but not over intersection. Specifically, for Wß X ß Y © K , WÐX Y Ñ œ ÐWX ÑY WÐX ∪ Y Ñ œ WX ∪ WY and ÐX ∪ Y ÑW œ X W ∪ Y W WÐX ∩ Y Ñ © WX ∩ WY and ÐX ∩ Y ÑW © X W ∩ Y W Also,

32

Fundamentals of Group Theory

+W œ +X

Í

WœX

Of course, we may generalize the set product to any nonempty finite collection W" ß á ß W8 of subsets of K by setting W" âW8 œ Ö=" â=8 ± =3 − W3 × On the other hand, if 5 is a positive integer, then it is customary to let W 5 œ Ö=5 ± = − W× Thus, in general, W # is a proper subset of the set product WW .

Subgroups The substructures of a group are defined as follows. Definition A nonempty subset L of a group K is a subgroup of K, denoted by L Ÿ K, if L is a group under the restricted product on K. If L Ÿ K and L Á K, we write L  K and say that L is a proper subgroup of K. If L" ß á ß L8 are subgroups of K, we write L" ß á ß L8 Ÿ K .… For example, ™ Ÿ , since ™ is an abelian group under addition. However, ™: is not a subgroup of ™, although it is a subset of ™ and it is a group as well: The issue is that ™: is not a group under ordinary addition of integers. However, if L is a subgroup under the first definition above, then L satisfies the second definition. To see this, multiplying the equation "L "L œ "L by the " inverse of "L in K gives "L œ "K . Thus, for all 2 − L , we have 22L œ " " " " œ 22K and so 2L œ 2K . There is another criterion for subgroups that involves checking only closure. Proof is left to the reader. Theorem 2.16 1) A nonempty subset \ of a group K is a subgroup of K if and only if \ is closed under the operations of taking inverses and products, that is, if and only if B−\

Ê

B" − \

and Bß C − \

Ê

BC − \

2) A nonempty finite subset \ of a group K is a subgroup of K if and only if it is closed under the taking of products.…

Groups and Subgroups

33

Theorem 2.17 The intersection of any nonempty family of subgroups of a group K is a subgroup of K.… Example 2.18 The set E8 of all even permutations in W8 is a subgroup of W8 . To see that E8 is closed under the product, if 5 and 7 are even, then they can each be written as a product of an even number of transpositions. Hence, 57 is also a product of an even number of transpositions and so is in E8 . The subgroup E8 is called the alternating subgroup of W8 . For 8   #, the alternating subgroup E8 has order 8xÎ#, that is, E8 is exactly half the size of W8 . To see this, note that 5 − W8 is odd if and only if Ð" #Ñ5 is even and 7 is even if and only if Ð" #Ñ7 is odd. Hence, the map 5 È Ð" #Ñ5 is a self-inverse bijection between E8 and the set of odd permutations.… A group has many important subgroups. One of the most important is the following. Definition The center ^ÐKÑ of a group K is the set of all elements of K that commute with all elements of K, that is, ^ÐKÑ œ Ö+ − K ± +, œ ,+ for all , − K× A group K is centerless if ^ÐKÑ œ Ö"×. A subgroup L of K is central if L is contained in the center of K.… Two subgroups of a group K can never be disjoint as sets, since each contains the identity of K . However, it will be very convenient to introduce the following terminology and notation. Definition Two subgroups L and O of a group K are essentially disjoint if L ∩ O œ Ö"× We introduce the notation L ìO to denote the set product of two essentially disjoint subgroups and refer to this as the essentially disjoint product of L and O .… Note that if 9ÐLÑ œ 8 and 9ÐOÑ œ 7, then 9ÐL ì OÑ œ 87

The Dedekind Law The following formula involving the intersection and set product is very handy and we will use it often. The simple proof is left to the reader.

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Fundamentals of Group Theory

Theorem 2.19 Let K be a group and let Eß Fß G Ÿ K with E Ÿ F . 1) (Dedekind law) EÐF ∩ GÑ œ F ∩ EG 2) E ∩ G œ F ∩ G and EG œ FG

Ê

EœF

Proof. We leave proof of the Dedekind law to the reader. For part 2), E œ EÐE ∩ GÑ œ EÐF ∩ GÑ œ F ∩ EG œ F ∩ FG œ F

…

We leave it to the reader to find an example to show that the condition that E Ÿ F is necessary in Dedekind's law, that is, EÐF ∩ GÑ is not necessarily equal to EF ∩ EG unless E Ÿ F .

Subgroup Generated by a Subset If K is a group and + − K, then the cyclic subgroup generated by + is the subgroup Ø+Ù œ Ö+5 ± 5 − ™× If 9Ð+Ñ œ 8  ∞, then +5 œ +5 mod 8 and so Ø+Ù œ Ö"ß +ß á ß +8" × Note that Ø+Ù is the smallest subgroup of K containing +, since any subgroup of K containing + must contain all powers of +. More generally, if \ is a nonempty subset of K, then the subgroup generated by \ , denoted by Ø\Ù, is defined to be the smallest subgroup of K containing \ and \ is called a generating set for Ø\Ù. Such a subgroup must exist; in fact, Theorem 2.17 implies that Ø\Ù is the intersection of all subgroups of K containing \ . The following theorem gives a very useful look at the elements of Ø\Ù. Theorem 2.20 Let \ be a nonempty subset of a group K and let \ " œ ÖB" ± B − \×. Let [ œ Ð\ ∪ \ " ч be the set of all words over the alphabet \ ∪ \ " . 1) If we interpret juxtaposition in [ as the group product in K and the empty word as the identity in K , then Ø\Ù œ [ and so Ø\Ù œ ÖB/"" âB/88 ± B3 − \ß /3 − ™ß 8  !× ∪ Ö"×

Groups and Subgroups

35

2) If K is abelian, then we can collect like factors and so Ø\Ù œ ÖB/"" âB/88 ± B3 − \ß B3 Á B4 for 3 Á 4ß /3 − ™ß 8  !× ∪ Ö"× Proof. It is clear that [ © Ø\Ù. It is also clear that [ is closed under product. /" 8 As to inverses, if A œ B/"" âB/88 − [ then A" œ B/ − [ . Hence, 8 âB" [ Ÿ Ø\Ù. However, since \ © [ , it follows that Ø\Ù Ÿ [ and so [ œ Ø\Ù.… Although the previous description of Ø\Ù is very useful, it does have one drawback: Distinct formal words in the set [ may be the same element of the group Ø\Ù, when juxtaposition in [ is interpreted as the group product. For instance, the distinct words B# B" and B are the same group element of Ø\Ù. We will discuss this issue in detail when we discuss free groups later in the book: The matter need not concern us further until then.

Finitely-Generated Groups A group K is finitely generated if K œ Ø\Ù for some finite set \ . If K has a generating set of size 8, then K is said to be 8-generated or to be an 8generator group.

The Burnside Problem There is a fascinating set of problems revolving around the following question. A finite group K is obviously finitely generated and periodic. In 1902, Burnside [39] asked about the converse: Is a finitely-generated periodic group finite? This is the general Burnside problem. A negative answer to the general Burnside problem took 62 years, when Golod [40] showed in 1964 that there are infinite groups that are $-generated and whose elements each have order a power of a fixed prime : (the power depending upon the element). However, this still leaves open some refinements of the general Burnside problem. For example, the Golod groups have elements of arbitrarily large order :8 , that is, they do not have finite exponent, as do finite groups. The Burnside problem is the problem of deciding, for finite integers 8 and 7, whether every 8-generated group of exponent 7 is finite. This problem has been the subject of a great deal of research since Burnside first formulated it in 1902. For example, it has been shown that there are infinite, finitely-generated groups of every odd exponent 7  ''& (Adjan [38], 1979) and of every exponent of the form #5 7, where 5   %) (Ivanov [42], 1992). The restricted Burnside problem, formulated in the 1930's, asks whether or not, for integers 8 and 7, there are a finite number (up to isomorphism) of finite 8-generated groups of exponent 7. In 1994, Zelmanov answered this question

36

Fundamentals of Group Theory

in the affirmative. For more on the Burnside problem, we refer the reader to the references located at the back of the book.

Subgroups of Finitely-Generated Groups A far simpler question related to finitely-generated groups is whether every subgroup of a finitely-generated group is finitely generated. We will show when we discuss free groups that for arbitrary groups this is false: There are finitelygenerated groups with subgroups that are not finitely generated. However, in the abelian case, this cannot happen. Theorem 2.21 Any subgroup of an 8-generated abelian group E is also 8generated. In particular, a subgroup of a cyclic group is cyclic. Proof. Let L Ÿ E. The proof is by induction on 8. If 8 œ ", then E œ Ø+Ù is cyclic. Let 5 be the smallest positive exponent for which +5 − L . Then Ø+5 Ù Ÿ L . However, if +7 − L , then 7 œ ;5  < where ! Ÿ <  5 and so +< œ +7;5 œ +7 Ð+5 Ñ; − L which can only happen if < œ !, whence +7 œ Ð+5 Ñ; − Ø+5 Ù. Thus, L œ Ø+5 Ù is also cyclic and so the result holds for 8 œ ". Assume the result is true for any group generated by fewer than 8   # elements. Let E œ ØB" ß á ß B8 Ù and let E8" œ ØB" ß á ß B8" Ù. By assumption, every subgroup of E8" is Ð8  "Ñ-generated, in particular, there exist 23 − L for which L ∩ E8" œ Ø2" ß á ß 28" Ù Now, every 2 − L has the form 2 œ +B/8 where + − E8" and / − ™. If / œ ! for all 2 − L , then L Ÿ E8" and the inductive hypothesis implies that L is at most Ð8  "Ñ-generated. So let us assume that / Á ! for some 2 − L and let = be the smallest positive integer for which 28 œ ,B=8 − L and , − E8" . For an arbitrary 2 œ +B/8 − L , where + − E8" , write / œ ;=  < where ! Ÿ <  =. Then 28; 2 œ Ð,B=8 Ñ; Ð+B/8 Ñ œ +, ; B/;= œ +, ; B<8 8 Since 28; 2 − L and +, ; − E8" , the minimality of = implies that < œ ! and so 28; 2 − E8" , that is, 2 − ØE8" ß 28 Ù. Hence, L © ØE8" ß 28 Ù œ Ø2" ß á ß 28" ß 28 Ù © L and so L œ Ø2" ß á ß 28" ß 28 Ù is 8-generated.…

Groups and Subgroups

37

The Lattice of Subgroups of a Group Let K be a group. The collection subÐKÑ of all subgroups of K is ordered by set inclusion. Moreover, subÐKÑ is closed under arbitrary intersection and has maximum element K. Hence, Theorem 1.7 implies that subÐKÑ is a complete lattice, where the meet of a family Y œ ÖL3 ± 3 − M× is the intersection 4Y œ 4 L3 œ , L3 3−M

3−M

and the join of Y is the smallest subgroup of K that contains all of the subgroups in Y , that is, 2Y œ 2 L3 œ , ÖW Ÿ K ± L3 Ÿ W for all 3× 3−M

It is also clear that the join of Y is the subgroup generated by the union of the L3 's. Specifically, since L3" œ L3 , it follows that the subgroup generated by Y is the set of all words over the union -L3 : 2 L3 œ 3−M

.L3

‡

œ Ö+3" â+38 ± +35 − L35 ß 8   !×

3−M

The join of Y is also denoted by ØY Ù and ØL3 ± 3 − MÙ. Note that, in general, the union of subgroups is not a subgroup. (The reader may find an example in the group of integers.) However, if L" Ÿ L# Ÿ â is an increasing sequence of subgroups of K (generally referred to as an ascending chain of subgroups), then it is easy to see that the union -L3 is a subgroup of K. More generally, the union of any directed family of subgroups is a subgroup. (A family W of subgroups of K is directed if for every Eß F − W, there is a G − W for which E Ÿ G and F Ÿ G .) Theorem 2.22 Let K be a group. Then subÐKÑ is a complete lattice, where meet is intersection and join is given by 2 L3 œ ¢. L3 £ 3−M

3−M

Also, subÐKÑ is closed under directed unions.…

Hasse Diagrams For a finite group of small order, we can sometimes describe the subgroup lattice structure using a Hasse diagram, which is a diagram of a partially ordered set that shows the covering relation.

38

Fundamentals of Group Theory

Example 2.23 The Hasse diagrams for the subgroup lattices of the group G% œ Ö"ß +ß +# ß +$ × and the %-group Z œ Ö"ß +ß ,ß +,× are shown in Figure 2.1.…

C4

V

{1,a2}

{1,a}

{1}

{1,b}

{1,ab}

{1}

S(C4)

S(V) Figure 2.1

Maximal and Minimal Subgroups of a Group The maximal and minimal subgroups of a group play an important role in the theory. At this point, we simply give the definitions. Definition Let K be a group and let L Ÿ K. 1) L is minimal if it is minimal in the partially ordered set of all nontrivial subgroups of K (under set inclusion). 2) L is maximal if it is maximal in the partially ordered set of all proper subgroups of K (under set inclusion).… We emphasize that the term maximal subgroup means maximal proper subgroup. Without the restriction to proper subgroups, K would be the only “maximal” subgroup of K. A similar statement can be made for minimal subgroups.

Subgroups and Conjugation Since the conjugate of a subgroup is also a subgroup, conjugation sends subÐKÑ to subÐKÑ. In fact, it is an order isomorphism of subÐKÑ. Theorem 2.24 Let K be a group and let + − K. The conjugation map #+ À subÐKÑ Ä subÐKÑ defined by #+ L œ L + is an order isomorphism on subÐKÑ. Hence, #+ preserves meet and join, that is, ’, L3 “ œ , L3+ +

and

’2 L3 “ œ 2 L3+ +

Groups and Subgroups

39

Proof. It is clear that EŸF

Í

E+ Ÿ F +

and so #+ is an order embedding of K into itself. But any subgroup E of K has the form E œ #+ Ð#+" EÑ and so #+ is also surjective.…

The Set Product of Subgroups If L and O are subgroups of a group K, then the set product LO is not necessarily a subgroup of K, as the next example shows. Example 2.25 In the symmetric group W$ , let L œ Ö+ß Ð" #Ñ×

and O œ Ö+ß Ð" $Ñ×

Then LO œ Ö+ß Ð" #Ñß Ð" $Ñß Ð" $ #Ñ× However, Ð" $ #Ñ# œ Ð" # $Ñ is not in LO and so LO is not a subgroup of W$ .… If LO is a subgroup of K, then since LO contains both L and O , we have OL © LO . Conversely, if OL © LO , then LO Ÿ K, since Ð25Ñ" œ 5 " 2" − OL © LO and Ð2" 5" ÑÐ2# 5# Ñ œ 2" Ð5" 2# Ñ5# œ 2" Ð2$ 5$ Ñ5# − LO where 23 − L and 53 − O . Thus, LO Ÿ K

Í

OL © LO

Moreover, if OL © LO , then equality must hold, since every B − LO has the form B œ Ð25Ñ" œ 5 " 2 " © OL for some 2 − L and 5 − O . Thus LO œ OL . Theorem 2.26 If Lß O Ÿ K, then the following are equivalent: 1) LO Ÿ K 2) OL © LO 3) OL œ LO , that is, L and O permute. In this case, LO œ L ” O is the join of L and O in subÐKÑ.…

The Size of L O There is a very handy formula for the size of the set product LO of two subgroups of a group K, which holds even if LO is not a subgroup of K. Consider the surjective map 0 À L ‚ O Ä LO defined by

40

Fundamentals of Group Theory

0 Ð2ß 5Ñ œ 25 The inverse map 0 " induces a partition c on L ‚ O whose blocks are the sets 0 " ÐBÑ for B − LO . Hence, there are kLO k blocks. To detemine the size of these blocks, let B œ 25 . Then any element of L ‚ O can be written in the form Ð2.ß /5Ñ for . − L and / − O and 0 Ð2.ß /5Ñ œ B

Í

2./5 œ 25

Í

./ œ "

Í

/ œ . "

and so 0 " ÐBÑ œ ÖÐ2.ß . " 5Ñ ± . − L ∩ O× Hence,

and so

¸0 " ÐBѸ œ kL ∩ O k kL kkO k œ kL ‚ O k œ kLO kkL ∩ O k

as cardinal numbers. Theorem 2.27 If K is a group and Lß O Ÿ K then

kL kkO k œ kLO kkL ∩ O k

as cardinal numbers. If L and O are finite subgroups, then kLO k œ

kL kkO k kL ∩ O k

…

The largest proper subgroups of a finite group K are the subgroups of order 9ÐKÑÎ#. The formula in Theorem 2.26 tells us something about how these large subgroups interact with other subgroups of K. Theorem 2.28 Let K be a finite group and let L Ÿ K have order 9ÐKÑÎ#. Then any subgroup W of K is either a subgroup of L or else kW ∩ L k œ kW kÎ#

In words, W lies either completely in L or half-in and half-out of L . Also, if + − W Ï L, then W œ ÐW ∩ LÑ “ +ÐW ∩ LÑ Proof. If W is not contained in L , then there is an + − W Ï L and so WL ª L “ +L But the latter has size 9ÐKÑ and so WL œ K . Then

Groups and Subgroups

41

kL kkW k œ kKkkL ∩ W k

implies that kW k œ #kL ∩ W k. The rest follows easily.…

Cosets and Lagrange's Theorem Let L Ÿ K. For + − K, the set +L is called a left coset of L in K. Similarly, the set L+ is called a right coset of L in K . The set of all left cosets of L in K is denoted by KÎL and the set of all right cosets is denoted LÏK . We will refer to left cosets simply as cosets, using the adjectives “left” and “right” only to avoid ambiguity. The map 0 À KÎL Ä LÏK defined by 0 Ð+LÑ œ L+" is easily seen to be a bijection and so kKÎL k œ kLÏKk

Since the multiplication map .+ À L Ä +L defined by .+ 2 œ +2 is a bijection, all cosets of a subgroup L have the same cardinality: k+L k œ kL k

To see that the distinct left cosets of L form a partition of K , we define an equivalence relation on K by +´,

if +L œ ,L

Now, +L œ ,L implies that , − +L . Conversely, if , − +L , then , œ +2 for some 2 − L and so ,L œ +2L œ +L . Hence, +´,

Í

+L œ ,L

Í

, − +L

and so the equivalence class containing + is precisely the coset +L . Thus, the distinct cosets KÎL form a partition of K . In particular, kKk œ kL k † kKÎL k

as cardinal numbers. When K is finite, this is the content of Lagrange's theorem. Theorem 2.29 Let K be a group and let L Ÿ K. 1) The set KÎL of distinct left cosets of L in K forms a partition of K , with associated equivalence relation satisfying +´,

Í

+L œ ,L

Í

, − +L

This equivalence relation is called equivalence modulo L and is denoted by + ´ , mod L

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Fundamentals of Group Theory

or just + ´ , when the subgroup is clear. Each element , − +L is called a coset representative for the coset +L , since ,L œ +L . 2) All cosets have the same cardinality and kKk œ kL k † kKÎL k

as cardinal numbers. 3) (Lagrange's theorem) If K is finite, then kKÎL k œ

kK k kL k

and so 9ÐLÑ ± 9ÐKÑ In particular, the order of an element + − K divides the order of K.… The converse of Lagrange's theorem fails: We will show later in the book that the alternating group E% has order "# but has no subgroup of order '. Note also that + ´ , mod L

, " + − L

Í

and so, in particular, + ´ , mod L

and

+ ´ , mod O

+ ´ , modÐL ∩ OÑ

Ê

Lagrange's Theorem and the Order of a Product Lagrange's theorem tells us something about the order of certain products 25 in a group even when 2 and 5 do not commute. Theorem 2.30 Let K be a group and let L and O be finite subgroups of K, with LO Ÿ K. 1) If 2 − L and 5 − O , then 9Ð25Ñ ± 9ÐLÑ9ÐOÑ In particular, if Ø2ÙØ5Ù Ÿ K is finite, then 9Ð25Ñ ± 9Ð2Ñ9Ð5Ñ 2) If R Ÿ K and 9ÐR Ñ is relatively prime to both 9ÐLÑ and 9ÐOÑ, then LR œ OR

Ê

L œO

Proof. For part 1), Lagrange's theorem implies that 9Ð25Ñ ± 9ÐLOÑ ± 9ÐLÑ9ÐOÑ

Groups and Subgroups

43

For part 2), if 2 − L then 2 œ 5+ where 5 − O and + − R . Hence 5 " 2 œ + − R and so 9Ð+Ñ ± 9ÐOÑ9ÐLÑ and 9Ð+Ñ ± 9ÐR Ñ, whence + œ ", that is, 2 œ 5 − O . Hence, L Ÿ O and a symmetric argument shows that L œ O.…

Euler's Formula The Euler phi function 9 is defined, for positive integers 8, by letting 9Ð8Ñ be the number of positive integers less than 8 and relatively prime to 8. The Euler phi function is important in group theory since the cyclic group G8 of order 8 has exactly 9Ð8Ñ generators. Theorem 2.31 (Properties of Euler's phi function) 1) The Euler phi function is multiplicative, that is, if 7 and 8 are relatively prime, then 9Ð78Ñ œ 9Ð7Ñ9Ð8Ñ 2) If : is a prime and 8   ", then 9Ð:8 Ñ œ :8" Ð:  "Ñ These two properties completely determine 9 . Proof. For part 1), consider the 7 ‚ 8 matrix Ô" Ö# Ö Ö EœÖ Ö< Ö

Õ7

7" 7#

#7  " #7  #

7<

#7  <

#7

$7

â Ð8  "Ñ7  " × â Ð8  "Ñ7  # Ù Ù ã Ù Ù â Ð8  "Ñ7  < Ù Ù ã Ø â 78

Note that the entries in the
44

Fundamentals of Group Theory

Ö: † "ß : † #ß á ß : † : 8" × which is a set of size :8" . Hence, 9Ð:8 Ñ œ :8  :8" .… Some simple group theory yields two famous old theorems from number theory. Theorem 2.32 (Euler's theorem) Let 8 be a positive integer. If + is an integer relatively prime to 8, then +9Ð8Ñ ´ " mod 8 This formula is called Euler's formula. Proof. Since ™‡8 is a multiplicative group of order 9Ð8Ñ, Lagrange's theorem implies that Euler's formula holds for + − ™‡8 and since Ð+  58Ñ9Ð8Ñ ´ +9Ð8Ñ ´ " mod 8 Euler's formula holds for all integers that are relatively prime to 8.… Corollary 2.33 (Fermat's little theorem) If : is a prime and Ð+ß :Ñ œ ", then +: ´ + mod :

…

Cyclic Groups Let us gather some facts about cyclic groups. Theorem 2.34 (Properties of cyclic groups) 1) (Prime order implies cyclic) Every group of prime order is cyclic. 2) (Smallest positive exponent) A finite abelian group K is cyclic if and only if minexpÐKÑ œ 9ÐKÑ. 3) (Subgroups) Every subgroup of a cyclic group is cyclic. a) (Lattice of subgroups: infinite case) If Ø+Ù is infinite, then each power +5 with 5   ! generates a distinct subgroup Ø+5 Ù and this accounts for all subgroups of Ø+Ù. b) (Lattice of subgroups: finite case) If 9Ð+Ñ œ 8 then for each . ± 8, the group Ø+Ù has exactly one subgroup W œ Ø+8Î. Ù of order . and exactly 9Ð.Ñ elements of order . , all of which lie in W . This accounts for all subgroups of Ø+Ù. It follows that for any positive integer 8, 8œ

9Ð.Ñ .±8

4) (Characterization by subgroups) If a finite group K of order 8 has the property that it has at most one subgroup of each order . ± 8, then K is cyclic (and therefore has exactly one subgroup of each order . ± 8).

Groups and Subgroups

45

5) (Direct products) A direct product K œ K" } â } K 7 of finite order is cyclic if and only if each K3 is cyclic and the orders of the factors K3 are pairwise relatively prime. Moreover, if ." ß á ß .8 are pairwise relatively prime positive integers, then the following hold: a) (Composition) If Ø+3 Ù is cyclic of order .3 , then Ø+" Ù } â } Ø+8 Ù œ ØÐ+" ß á ß +8 ÑÙ b) (Decomposition) If K œ Ø+Ù has order . œ ." â.8 , then Ø+Ù œ Ø+" ÙâØ+8 Ù where 9Ð+3 Ñ œ .3 and

Ø+3 Ù ∩ $Ø+4 Ù œ Ö"× 4Á3

for all 3. Proof. Part 2) follows from Theorem 2.12. For part 3b), the generators of the cyclic groups of order . are the elements of order .. However, Theorem 2.9 implies that the elements of K of order . are Ö+<Ð8Î.Ñ ± Ð<ß .Ñ œ "× and these all lie in the one cyclic subgroup Ø+8Î. Ù of order . . Hence, there can be no other cyclic subgroups of order .. For the final statement, the sum .±8 9Ð.Ñ simply counts the elements of K by their order. To prove 4), let H be the set of all orders of elements of K. If + − K has order 9Ð+Ñ œ . , then Ø+Ù is the unique subgroup in K of order . and so all elements of order . must be in Ø+Ù. It follows that there are exactly 9Ð.Ñ elements in K of order . − H. Hence, 8œ

9Ð.Ñ Ÿ .−H

9Ð.Ñ œ 8 .±8

and so equality holds, whence H is the set of all divisors of 8. In particular, 8 − H and so K is cyclic. For part 5), if each K3 œ Ø+3 Ù is cyclic of order .3 , where the .3 's are pairwise relatively prime, then 9ÐÐ+" ß á ß +8 ÑÑ œ lcmÐ." ß á ß .8 Ñ œ $.3 œ 9ÐKÑ

and so K œ ØÐ+" ß á ß +8 ÑÙ is cyclic. (This also proves part 5a).) Conversely, suppose that K is cyclic and for each +3 − K3 , let

46

Fundamentals of Group Theory

+3 œ Ð"ß á ß "ß +3 ß "ß á ß "Ñ s where +3 is in the 3th position. The subgroups KÐ3Ñ œ Ö"× } â } Ö"× } K3 } Ö"× } â } Ö"× where K3 is in the 3th position are cyclic and if KÐ3Ñ œ Ø+ s3 Ù has order .3 , then K3 œ Ø+3 Ù is also cyclic of order .3 . Moreover, $.3 œ 9ÐKÑ œ minexpÐKÑ œ lcmÐ." ß á ß .8 Ñ

and so the orders .3 are pairwise relatively prime. For part 5b), since K œ Ø+Ù is abelian, the product T œ Ø+" ÙâØ+8 Ù is a subgroup of K and since +" â+8 − T has order 9Ð+" â+8 Ñ œ lcmÐ." ß á ß .8 Ñ œ . œ 9ÐØ+ÙÑ it follows that T œ K. Finally, if

α − Ø+3 Ù ∩ $Ø+4 Ù 4Á3

then 9ÐαÑ divides .3 as well as the product #4Á3 .4 , which are relatively prime and so α œ ".…

Homomorphisms of Groups We will discuss the structure-preserving maps between groups in detail in a later chapter, but we wish to introduce a few definitions here for immediate use. Definition Let K and L be groups. A function 5À K Ä L is called a group homomorphism (or just homomorphism) if 5Ð+,Ñ œ Ð5+ÑÐ5,Ñ for all +ß , − K. A bijective homomorphism is an isomorphism. When 5À K Ä L is an isomorphism, we write 5À K ¸ L or simply K ¸ L and say that K and L are isomorphic.… A property of a group is isomorphism invariant if whenever a group K has this property, then so do all groups isomorphic to K. For example, the properties of being finite, abelian and cyclic are isomorphism invariant.

More Groups Let us look at a few classes of groups. As we have seen, some groups are given names, for example, the cyclic groups, the symmetric groups, the quaternion group and the dihedral groups (to be defined below). Actually, these are really isomorphism classes of groups. For instance, if K ¸ W8 , then we might

Groups and Subgroups

47

reasonably refer to K as a symmetric group as well. After all, it is the algebraic structure that is important and not the labeling of the elements of the underlying set.

Cyclic Groups Theorem 2.33 describes the subgroup lattice structure of a cyclic group. Let us take a somewhat closer look at this structure. Let G8 Ð+Ñ be cyclic of finite order 8 and let H8 be the lattice (under division) of positive integers less than or equal to 8 that divide 8. Then the map 5À H8 Ä subÐG8 Ð+ÑÑ defined by 5. œ G. œ Ø+8Î. Ù is an order isomorphism from H8 to subÐG8 Ð+ÑÑ, since 5 is surjective and ." ± .#

Í

Ð8Î.# Ñ ± Ð8Î." Ñ

Í

G." Ÿ G.#

Thus, subÐG8 Ð+ÑÑ is order isomorphic to H8 . For example, Figure 2.2 shows the lattices H#% and subÐG#% Ð+ÑÑ.

24

G

8

12

C8

C12

4

6

C4

C6

2

3

C2

C3

1

{1} Figure 2.2

For the infinite case, we must settle for an order anti-isomorphism. Let ™ be the lattice of nonnegative integers under division. If K œ Ø+Ù is an infinite cyclic group, then the map 5À ™ Ä subÐKÑ defined by 55 œ Ø+5 Ù is an order antiisomorphism, since 5 is surjective and 5±8

Í

Ø+8 Ù Ÿ Ø+5 Ù

Thus, subÐKÑ is order anti-isomorphic to ™ .

The Quaternion Group Let KPÐ#ß ‚Ñ be the general linear group of all nonsingular # ‚ # matrices over the complex numbers. Let " M œŒ !

! 3 ßE œ Œ " !

! ! ßF œ Œ 3 "

" ! ßG œ Œ ! 3

Then it is easy to see that E# œ F # œ G # œ EFG œ M

3 !

48

Fundamentals of Group Theory

and ÐMÑ\ œ \ÐMÑ œ \ for \ œ Eß F or G . These equations are sufficient to determine all products in the set W œ Ö „ Mß „ Eß „ Fß „ G× In particular, EF œ Gß

FG œ Eß

EG œ EEF œ F

and FE œ FFG œ G GF œ EFF œ E GE œ EFFG œ EG œ F Thus, W is a subgroup of KPÐ#ß ‚Ñ of order ). Any group isomorphic to W is called a quaternion group. Note that we have defined the quaternion group in such a way that it is clearly a group, since we have defined it as a subgroup W of the known group KPÐ#ß ‚Ñ. However, the quaternion group is often defined without mention of matrices. In this case, it becomes necessary to verify that the definition does indeed constitute a group. On the other hand, we can leverage our knowledge of existing groups (in particular W ) by the following simple device: A bijection 0 À K Ä \ from a group K to a set \ can be used to transfer the group product from K to \ by setting 0 Ð+Ñ0 Ð,Ñ œ 0 Ð-Ñ

if +, œ -

This makes \ a group and 0 an isomorphism from K to \ . Now, the quaternion group is often defined as the set U œ Ö"ß 3ß 4ß 5ß "ß 3ß 4ß 5× with multiplication defined so that " is the identity and 3# œ 4# œ 5 # œ 345 œ " Ð"ÑB œ BÐ"Ñ œ B

(2.35)

for B œ 3ß 4ß 5 . As with the set W defined above, these rules are sufficient to define all products of elements of U. Rather than show directly that this forms a group, that is, rather than verifying directly the associative, identity and inverse properties, we can simply observe that the map 0 À W Ä U defined by

Groups and Subgroups

49

0 ÐMÑ œ "ß 0 ÐMÑ œ " 0 ÐEÑ œ 3ß 0 ÐEÑ œ 3 0 ÐFÑ œ 4ß 0 ÐFÑ œ 4 0 ÐGÑ œ 5ß 0 ÐGÑ œ 5 is a bijection and that 0 Ð\Ñ0 Ð] Ñ œ 0 Ð^Ñ

Í

\] œ ^

for all \ß ] ß ^ − W . Hence, the product in U is the image of the product in W and so U is a group because W is a group. Note that the main savings here is in the fact that we do not need to verify that the product in U is associative. Of course, someone had to verify that matrix multiplication is associative, and we do appreciate that effort very much. Equations (2.35) are equivalent to 3# œ 4# œ 5 # œ " 34 œ 5ß 45 œ 3ß 53 œ 4 Ð"ÑB œ BÐ"Ñ œ B for B œ 3ß 4ß 5 and many authors use these equations to define the quaternion group. In order to describe the quaternion group, it is not necessary to mention explicitly all four elements ", 3, 4 and 5 , since " œ 3# and 5 œ 34. In fact, U can also be defined as the group satisfying the following conditions: U œ Ø3ß 4Ùß

9ÐUÑ œ )ß

3% œ "ß

3# œ 4 # ß

43 œ 3$ 4

To see this, if " ³ 3# ,

5 ³ 34 and

B ³ Ð"ÑB

for B œ "ß 3ß 4ß 5 , then U œ Ö"ß 3ß 3# ß 3$ ß 4ß 34ß 3# 4ß 3$ 4× œ Ö"ß 3ß "ß 3ß 4ß 5ß 4ß 5× Moreover, since 43 œ 3$ 4, it follows that 5 # œ 3434 œ 33$ 44 œ " and 345 œ 3434 œ 33$ 44 œ 4# œ " Note that the conditions

50

Fundamentals of Group Theory

U œ Ø3ß 4Ùß

3 Á 4ß

9Ð3Ñ œ %ß

3# œ 4 # ß

343 œ 4

imply that 9ÐKÑ œ ) and so they provide perhaps the most succinct definition of the quaternion group. Subgroups of the Quaternion Group The quaternion group has a simple subgroup lattice. We leave it as an exercise to verify that the Hasse diagram for subÐUÑ is given by Figure 2.3. The subgroup Ø"Ù has order # and the other nontrivial proper subgroups have order %.

Q

<j>



<-1> {1} Figure 2.3

The Dihedral Groups Many groups come from geometry. Here is one of the most famous. First, we need a bit of terminology. A rigid motion of the plane is a bijective distancepreserving map of the plane. If T is a nonempty subset of the plane, then a symmetry of T is a rigid motion of the plane that sends T onto itself. Now, for 8   #, let T be a regular 8-gon in the plane, whose center passes through the origin. Figure 2.4 shows the cases 8 œ % and 8 œ &.

1

1 5

4

2

2 4

3

3

Figure 2.4 (For 8 œ #, T is a line segment.) Label the vertices "ß á ß 8 in clockwise order, with vertex " on the positive vertical axis. Let Z be the set of vertices of T .

Groups and Subgroups

51

The symmetry group K8 of T consists of all symmetries of T . It is possible to prove that the symmetries of T are the same as the symmetries of the vertex set Z and so we may regard K8 as the set of all symmetries of Z . In fact, each symmetry of Z is a permutation of Z and is uniquely determined by that permutation. Indeed, it is customary to think of the elements of K8 as permutations of the labeling set M8 œ Ö"ß á ß 8×, that is, as elements of W8 . Here are a few simple facts concerning K8 , for 8   #. 1) Since any $ − K8 preserves adjacency, if $ B œ C then $ ÐB  "Ñ œ C  " or $ÐB  "Ñ œ C  ". We denote this by writing $À ÐBß B  "Ñ È ÐCß C  "Ñ or $À ÐBß B  "Ñ È ÐCß C  "Ñ In the former case, we say that $ preserves orientation in the pair ÐBß B  "Ñ and in the latter case, $ reverses orientation. Note that addition and subtraction are performed modulo 8, but then ! is replaced by 8. Hence, for example, "  " œ 8 and so " and 8 are adjacent. 2) Any $ − K8 is uniquely determined by its value on two adjacent elements ÐBß B  "Ñ of M8 . To see this, we may assume that 8   $. If $ preserves orientation on ÐBß B  "Ñ, that is, if $À ÐBß B  "Ñ È ÐCß C  "Ñ then since $ ÐB  #Ñ is adjacent to $ ÐB  "Ñ œ C  ", we have $ ÐB  #Ñ œ C or $ ÐB  #Ñ œ C  #. But $ ÐB  #Ñ Á $ B œ C and so $À ÐB  "ß B  #Ñ È ÐC  "ß C  #Ñ Repeating this argument gives $À ÐB  5ß B  5  "Ñ È ÐC  5ß C  5  "Ñ for all 5 . A similar argument holds if $ reverses orientation, showing that $À ÐB  5ß B  5  "Ñ È ÐC  5ß C  5  "Ñ Note also that if $ preserves orientation for one adjacent pair, then it preserves orientation for all adjacent pairs and so we can simply say that $ either preserves orientation or reverses orientation. The group K8 contains both the 8-cycle 3 œ Ð" # â 8Ñ which represents a clockwise rotation through $'!Î8 degrees, and the permutation 5 that represents a reflection across the vertical axis. If 8 is even,

52

Fundamentals of Group Theory

then

5 œ Ð# 8ÑÐ$ 8  "ÑâŠ

8 8  #‹ # #

and if 8 is odd then 5 œ Ð# 8ÑÐ$ 8  "ÑâŒ

8" 8" " # #

To see that K8 is generated by the rotation 3 and the reflection 5 , let $ − K8 . If $ preserves orientation, then $À Ð8ß "Ñ È Ð5ß 5  "Ñ and so $ œ 35 − Ø3ß 5Ù. On the other hand, if $ reverses orientation, then $À Ð8ß "Ñ È Ð5ß 5  "Ñ and so the following hold: $5À Ð#ß "Ñ È Ð5ß 5  "Ñ $5À Ð"ß #Ñ È Ð5  "ß 5Ñ $5À Ð8ß "Ñ È Ð5  #ß 5  "Ñ Hence, $5 œ 35" and so $ œ 35" 5 − Ø3ß 5Ù. Thus K8 œ Ø5ß 3Ù. To examine the group structure of K8 , we have 9Ð3Ñ œ 8

and 9Ð5Ñ œ #

Also, 5353À Ð"ß 8Ñ È Ð"ß 8Ñ and so 53 is an involution, which gives the commutativity rule 35 œ 53" œ 538" and so every element of K8 can be written in the form 5/ 35 . Thus, K8 can be described succinctly by K8 œ Ø5ß 3Ùß

9Ð3Ñ œ 8ß

9Ð5Ñ œ 9Ð53Ñ œ #

It is easy to see that the #8 elements Ö+ß 3ß á ß 38" ß 5ß 53ß á ß 538" × are distinct and so 9ÐK8 Ñ œ #8. Another description of K8 can be gleaned from the fact that Ø5ß 3Ù œ Ø5ß 53Ù and so K8 is generated by a pair of involutions whose product has finite order, that is, if 1 œ 53, then

Groups and Subgroups

K8 œ Ø5ß 1Ùß

9Ð5Ñ œ 9Ð1Ñ œ #ß

53

9Ð51Ñ œ 8

Without reference to geometry, a finite group K is a dihedral group if K is isomorphic to the symmetry group K8 , for some 8   #. Thus, a group K is a dihedral group if any of the following equivalent descriptions hold (here 3 and 5 are just symbols): 1) (Common description) K œ Ö" œ 3! ß 3ß á ß 38" ß 5ß 53ß á ß 538" × has order #8 and 38 œ "ß

5# œ "

and 35 œ 538"

2) (Succinct description 1) K œ Ø5ß 3Ùß

9Ð3Ñ œ 8   #

and 9Ð5Ñ œ 9Ð53Ñ œ #

3) (Succinct description 2) K œ Ø5ß 1Ùß

9Ð5Ñ œ 9Ð1Ñ œ #ß

9Ð51Ñ œ 8   #

Note that Ð535 ÑÐ535 Ñ œ 5535 35 œ " and so all elements of the form 535 are involutions. Unfortunately, the dihedral group K of order #8 is denoted by H8 by some authors (reflecting the fact that K consists of symmetries of 8 vertices) and by H#8 by other authors (reflecting the fact that K is a group of order #8). We will use the notation H#8 . Also, in view of the development of H#8 , we will often refer to 3 as a rotation and 5 as a reflection, even if 3 and 5 are not actually maps. For 8 œ ", the reflection 5 is the identity and the rotation 3 is the transposition Ð" #Ñ and so H# œ Ö+ß Ð" #Ñ× is the cyclic group G# . For 8 œ #, the dihedral group H% is G# } G# . For 8   $, we have proved that the dihedral groups exist, namely, as certain subgroups of W8 . Note that H#8 is nonabelian for 8   $. Ubiquity of the Dihedral Group Succinct description 2 of the dihedral group shows that dihedral groups occurs quite often and reinforces the idea that a dihedral group need not consist specifically of symmetries of the plane. Theorem 2.36 Let K be a group. If +ß , − K are distinct involutions for which 9Ð+,Ñ œ 8  ∞, then the subgroup Ø+ß ,Ù is dihedral of order #8.…

54

Fundamentals of Group Theory

Subgroups of the Dihedral Groups Let us determine the subgroups of H#8 . If W Ÿ H#8 , then Theorem 2.28 implies that there are two possibilities. If W Ÿ Ø3Ù, then W œ Ø38Î. Ù for some . ± 8. Otherwise, W ∩ Ø3Ù œ Ø38Î. Ù for some . ± 8 and if 5 is the smallest positive integer for which 535 − W , then W œ ÐW ∩ Ø3ÙÑ “ 535 ÐW ∩ Ø3ÙÑ œ Ø38Î. Ù “ 535 Ø38Î. Ù œ Ø535 ß 38Î. Ù Note that since 535 and 535 38Î. œ 5358Î. are involutions and 9Ð38Î. Ñ œ ., it follows that W is dihedral of order #., generated by the “reflection” 535 and the “rotation” 38Î. . Thus, the subgroups of H#8 fall into two categories: subgroups of Ø3Ù, which are cyclic and the rest, which are dihedral. Also, for distinct values of 5 in the range ! Ÿ 5  8Î. , the sets W.ß5 œ 535 Ø38Î. Ù “ Ø38Î. Ù œ Ø535 ß 38Î. Ù are distinct subgroups of H#8 and this accounts for all of the dihedral subgroups of H#8 . We can now summarize. Theorem 2.37 The subgroups of the dihedral group H#8 are of two types. For each . ± 8, we have 1) the cyclic subgroup Ø38Î. Ù of order . and 2) for each ! Ÿ 5  8Î. , the dihedral subgroup W.ß5 œ Ø535 ß 38Î. Ù œ 535 Ø38Î. Ù “ Ø38Î. Ù of order #..…

The Symmetric Groups The lattice of subgroups of the symmetric group W8 is rather complicated. Indeed, a famous theorem of Arthur Cayley [7] from 1854 (to be discussed in detail later), says that for any group K, the symmetric group WK contains a subgroup that is an “exact copy” of K. Thus, a complete description of the subgroup lattice of the symmetric groups would constitute a complete description of all groups, which does not yet exist!

The Additive Rationals Let us examine the subgroup lattice of the additive group  of rational numbers. Let ™ denote the set of positive integers. We say that +Î, −  is reduced if + and , are relatively prime and ,  !. Let L Ÿ  be nontrivial. Let M be the set of integers in L , let R be the set of numerators of the reduced elements of L and let H be the set of positive denominators of the reduced elements of L .

Groups and Subgroups

55

If + − R , then +Î, − L for some integer , and so + − M . Thus, M œ R . Moreover, if 3 is the smallest positive integer in M , then every + − M is an integral multiple of 3, for if + œ ;3  < where ! Ÿ <  3, then < œ +  ;3 − M and the minimality of 3 implies that < œ !. Thus, R œ M œ ™3. If +3Î, − L is reduced, then there exist integers ? and @ for which ?+  @, œ ", whence 3 Ð?+  @,Ñ3 ?+3 œ œ  @3 − L , , , Thus, +3 −L ,

Í

3 −L ,

for +ß , − ™, ,  !. Thus, L œœ

+3 ¹ + − ™ß , − H ,

It follows that H is closed under products, since if 3Î, and 3Î- in L are reduced, then 3# Î,- is also reduced and in L . Also, if . − H and / ± . , then / − H. Thus, H is closed under factors as well. It follows that we can describe H by describing which prime powers lie in H. If : is prime, let H: be the set of all powers of : that lie in H. Then H: has one of three forms. 1) If : ± 3, then H: œ Ö"× since if : − H then 3Î: − L is an integer, contradicting the fact that 3 is the smallest positive integer in L . 2) If : y± 3 and there is a largest integer 7Ð:Ñ for which :7Ð:Ñ − H, then H: œ Ö"ß á ß :7Ð:Ñ × 3) If : y± 3 and there is no largest integer 7Ð:Ñ for which :7Ð:Ñ − H, then H: œ Ö"ß :ß :# ß á × In case 1) we set 7Ð:Ñ œ ! and in case 3) we set 7Ð:Ñ œ ∞ and so 7Ð:Ñ is defined for all primes. Let :" ß :# ß á be the sequence of all primes and let 7ÐLÑ œ Ð7Ð:" Ñß 7Ð:# Ñß á Ñ For convenience, we say that a sequence Ð+" ß +# ß á Ñ where +3 is a nonnegative integer or +3 œ ∞ is acceptable for 3 if :5 ± 3 implies that +5 œ !. Thus, 7ÐLÑ is acceptable for 3.

56

Fundamentals of Group Theory

Let us adopt the notation Ð+" ß +# ß á Ñ Ÿ Ð," ß ,# ß á Ñ to mean that +5 Ÿ ,5 for all 5 . If , − H, we may write , œ :"/" :#/# â with the understanding that all but a finite number of exponents /5 are zero. Let /Ð,Ñ œ Ð/" ß /# ß á Ñ Then for any positive integer , , ,−H

Í

/Ð,Ñ Ÿ 7ÐLÑ

and so the pair Ð3ß 7ÐLÑÑ completely determines L since L œœ

+3  º+ − ™ß , − ™ ß /Ð,Ñ Ÿ 7ÐLÑ ,

On the other hand, let 3 − ™ and let = œ Ð7" ß 7# ß á Ñ be acceptable for 3. Then the set LÐ3ß=Ñ œ œ

+3  º + − ™ß , − ™ ß /Ð,Ñ Ÿ = ,

is a nontrivial subgroup of  for which 7ÐLÑ œ =. It is clear that L is closed under negatives. Also, if +3Î, − L and -3Î. − L are reduced, then +3 -3 ?3  œ −L , . lcmÐ,ß .Ñ is reduced and since /ÐlcmÐ,ß .ÑÑ Ÿ = it follows that L is closed under addition. Thus, the subgroups LÐ3ß=Ñ of  correspond bijectively to the pairs Ð3ß =Ñ, where 3 − ™ and where = œ Ð7" ß 7# ß á Ñ is acceptable for 3.

*An Historical Perspective: Galois-Style Groups Let us conclude this chapter with a brief historical look at groups. Evariste Galois (1811–1832) was the first to develop the concept of a group, in connection with his research into the solutions of polynomial equations. However, Galois' version of a group is quite different from the modern version we see today. Here is a brief look at groups as Galois saw them (using a bit more modern terminology than Galois used).

Groups and Subgroups

57

Consider a table in which each row contains an ordered arrangement of a set \ of distinct symbols (such as the roots of a polynomial), for example + , + ,

, + , + -

, + , +

. . . / / /

/ / / . . .

where \ œ Ö+ß ,ß -ß .ß /×. Then each pair of rows defines a permutation of \ , that is, a bijective function on \ . Galois considered tables of ordered arrangements with the property that the set E3 of permutations that transform a given row <3 into the other rows (or into itself) is the same for all rows <3 , that is, E3 œ E4 for all 3ß 4. Let us refer to this type of table, or list of ordered arrangements, as a Galois-style group. In modern terms, it is not hard to show that a list of ordered arrangements is a Galois-style group if and only if the corresponding set E ( œ E3 ) of permutations is a subgroup of the symmetric group W\ . To see this, let the permutation that transforms row <3 to row <4 be 13ß4 . Then Galois' assumption is that the sets E3 œ Ö13ß" ß á ß 13ß8 × are the same for all 3. This implies that for each 3ß ? and 4, there is a @ for which 13ß? œ 14ß@ . Hence, 13ß? 13ß4 œ 14ß@ 13ß4 œ 13ß@ − E3 and so E3 is closed under composition and is therefore a group. Conversely, if E" is a permutation group, then since 13ß4 œ 1"ß4 13ß" œ 1"ß4 Ð1"ß3 Ñ" − E" it follows that E3 œ E" for all 3 and so the ordered arrangement that corresponds to E" is a Galois-style group. Galois appears not to be entirely clear about a precise meaning of the term group, but for the most part, he uses the term for what we are calling a Galoisstyle group. Galois also worked with subgroups and recognized the importance of what we now call normal subgroups (defined in the next chapter), although his definition is quite different from what we would see today. When Galois' work was finally published in 1846, fourteen years after he met his untimely death in a duel at the age of 21, the theory of finite permutation groups had already been formalized by Augustin Louis Cauchy (1789–1857),

58

Fundamentals of Group Theory

who likewise required only closure under product, but who clearly recognized the importance of the other axioms by introducing notations for the identity and for inverses. Arthur Cayley (1821–1895) was the first to consider, in 1854, the possibility of more abstract groups and the need to axiomatize associativity. He also axiomatized the identity property, but still assumed that each group was a finite set and so had no need to axiomatize inverses (only the validity of cancellation). It was not until 1883 that Walther Franz Anton von Dyck (1856–1934), in studying the relationship between groups and geometry, made explicit mention of inverses. It is also interesting to note that Cayley's famous theorem (to be discussed in Chapter 4), to the effect that every group is isomorphic to a permutation group, completes a full circle back to Galois (at least for finite groups)!

Exercises 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14.

Let K be a group. a) Prove that K has exactly one identity. b) Prove that each element has exactly one inverse. Prove that any group K in which every nonidentity element has order # is abelian. Let L Ÿ K and let 1 − K have order 8. Show that if 15 − L for Ð5ß 8Ñ œ " then 1 − L . Show that a finite subset W of a group K is a subgroup if and only if it is closed under products. Show that if +ß , − K then 9Ð+,Ñ œ 9Ð,+Ñ. Let K be a group with center ^  K. Prove that if every element of K that is not in ^ has finite order, then K is periodic. Show that the center of U is equal to Ö"ß "×. Show that in a group of even order, there is an element of order #. (Do not use Cauchy's theorem, if you know it.) Hint: such an element is equal to its own inverse. Find an example of a group K and three distinct primes :ß ; and < for which K has elements + and , satisfying 9Ð+Ñ œ : , 9Ð,Ñ œ ; and 9Ð+,Ñ œ <. Prove that a group with only finitely many subgroups must be finite. Let K be a finite group of order 7 and let Ð8ß 7Ñ œ ". Show that every element 1 of K has a unique 8th root 2, where 28 œ 1. Prove that the group  of rational number has no minimal subgroups. a) Find a group K and subgroups L and O for which LO is not a subgroup of K. b) If K œ LOP where Lß O and P are subgroups of K, does it follow that Lß O and P commute (under set product)? Let J be a field and let J ‡ be the multiplicative group of nonzero elements of J .

Groups and Subgroups

59

a)

If J is a finite field, show that J ‡ is cyclic. Hint: Use the fact that a polynomial equation of degree 8 has at most 8 distinct solutions in J . b) Prove that if J is an infinite field, then J ‡ is not cyclic. Hint: What are the orders of the nonidentity elements? 15. Let W be a nonempty set that has an associative binary operation, denoted by juxtaposition. Show that W is a group if and only if +W œ W œ W+ for all + − W . 16. Let K be a nonempty set with an associative binary operation. Assume that there is a left identity "P , that is, "P + œ + for all + − K and that each element + has a left inverse +P , that is, +P + œ "P . Prove that K is a group under this operation and that "P œ " and +P œ +" . 17. Let K be a finite abelian group of order 8. Show that the product of all of the elements of K is equal to the product of all involutions in K (or " if K has no involutions). Apply this to the multiplicative group ™‡: where : is prime to deduce that Ð:  "Ñx ´ " mod : which is known as Wilson's theorem. 18. Draw a Hasse diagram of the subgroup lattice of a) the symmetric group W$ b) the dihedral group H) . 19. (Dihedral group) Show that 535 is the same as counterclockwise rotation 3" . 20. (Dihedral group) Let T be a regular #8-gon. Show that we get the same dihedral group if we use an axis of symmetry that goes through two vertices or through the midpoint of opposite sides of T . 21. (Dihedral group) Find the center of the dihedral group H#8 . 22. Let K be a group and suppose that +ß , − K satisfy +# œ " and +,# + œ ,$ . Prove that , & œ ". 23. Let  be the additive group of rational numbers. Let Ð3ß Ð85 ÑÑ describe L Ÿ  and let Ð4ß Ð75 ÑÑ describe O Ÿ . a) Under what conditions is O Ÿ L ? b) Under what conditions is L cyclic? 24. Let K be a finite group and W and X be subsets of K. Prove that either kW k  kX k Ÿ kKk or K œ WX . 25. A group K is locally finite if every finitely-generated subgroup is finite. a) Prove that a locally finite group is periodic. b) Prove that if K is abelian and periodic, then it is locally finite. 26. A group K is said to be locally cyclic if every finitely-generated subgroup of K is cyclic. a) Prove that K is locally cyclic if and only if every pair of elements of K generates a cyclic subgroup. b) Prove that a locally cyclic group is abelian.

60

Fundamentals of Group Theory c) d) e) f)

Prove that any subgroup of a locally cyclic group is locally cyclic. Prove that a finitely-generated locally cyclic group is cyclic. Find an example of a finitely-generated group that is not locally cyclic. Show that the subgroup K of the nonzero complex numbers (under multiplication) defined by K œ š/#138Î: ± 8ß 5 − ™› 5

where : is a fixed prime is locally cyclic but not cyclic. g) Prove that if K is locally cyclic, then all nonidentity elements have infinite order or else all elements have finite order. h) Prove that the additive group of rational numbers is locally cyclic. i) Show that any locally cyclic group whose nonidentity elements have infinite order is isomorphic to a subgroup of the additive group . j) The distributive laws are E ” ÐF ∩ GÑ œ ÐE ” FÑ ∩ ÐE ” GÑ E ∩ ÐF ” GÑ œ ÐE ∩ FÑ ” ÐE ∩ GÑ for Eß Fß G Ÿ K. Prove that each distributive law implies the other. A lattice that satisfies the distributive laws is said to be a distributive lattice. (It is possible to prove that subÐKÑ is a distributive lattice if and only if K is locally cyclic. This is difficult. A complete solution can be found in Marshall Hall's book The Theory of Groups [16].)

Ascending Chain Condition A group K satisfies the ascending chain condition (ACC) on subgroups if every ascending sequence L" Ÿ L# Ÿ â of subgroups must eventually be constant, that is, if there is an 8  ! such that L85 œ L8 for all 5  !. 27. A group K satisfies the maximal condition on subgroups if every nonempty collection of subgroups has a maximal member. Prove that a group K satisfies the maximal condition on subgroups if and only if it satisfies the ascending chain condition on subgroups. 28. A group K satisfies the ACC on subgroups if and only if every subgroup of K is finitely generated.

Chapter 3

Cosets, Index and Normal Subgroups

We begin this chapter with a more careful look at subgroups, cosets and indices.

Cosets and Index The number of cosets of a subgroup plays an important role in group theory. Definition Let K be a group. The index of L Ÿ K , denoted by ÐK À LÑ, is the cardinality of the set KÎL of all distinct left cosets of L in K , that is, ÐK À LÑ œ kKÎL k

…

Recall that kLÏKk œ kKÎL k and so the index is also the cardinality of the set of right cosets of L in K. It is convenient to extend the quotient and index notation as follows: If L Ÿ K and if \ is any nonempty subset of K, then we write \ÎL œ ÖBL ± B − \×

and denote the cardinality of \ÎL by Ð\ À LÑ. This is not entirely standard notation, but it is useful. For example, if O is a subgroup of K , but LO is not a subgroup, we may still want to consider the set LÎO œ LOÎO and the index ÐLO À OÑ. We will also have use for the following concept. Definition Let L Ÿ K. 1) A set consisting of exactly one element from each coset in KÎL is called a left transversal for L in K (or for KÎL ). 2) A set consisting of exactly one element from each right coset in LÏK is called a right transversal for L in K (or for LÏK).… Now we can give some important properties of the index.

S. Roman, Fundamentals of Group Theory: An Advanced Approach, DOI 10.1007/978-0-8176-8301-6_3, © Springer Science+Business Media, LLC 2012

61

62

Fundamentals of Group Theory

Theorem 3.1 Let K be a group. 1) If L Ÿ K, then ÐK À LÑ œ "

Í

KœL

2) If \ © K is a union of cosets of O Ÿ K, then

k\ k œ kO k † Ð\ À OÑ

Hence, if Lß O Ÿ K, then

kLO k œ kO k † ÐLO À OÑ

In particular,

kKk œ kO k † ÐK À OÑ

and so if K is finite, then ÐK À OÑ œ

kK k kO k

3) (Multiplicativity) If L Ÿ O Ÿ K then ÐK À LÑ œ ÐK À OÑÐO À LÑ as cardinal numbers. Hence, L ŸO

and ÐK À LÑ œ ÐK À OÑ  ∞

Ê

L œO

4) Let Lß O Ÿ K. If K is finite or if LO Ÿ K , then ÐK À OÑ œ ÐLO À OÑ  ∞

Ê

K œ LO

5) If Lß O Ÿ K, then ÐLO À OÑ œ ÐL À L ∩ OÑ 6) (Poincaré's theorem) Let L" ß á ß L8 Ÿ K and ÐK À L3 Ñ  ∞ for all 3 and let M5 œ L" ∩ â ∩ L5 . Then Poincaré's inequality holds: ÐK À L" ∩ â ∩ L8 Ñ Ÿ ÐK À L" ÑâÐK À L8 Ñ and so, in particular, ÐK À L" ∩ â ∩ L8 Ñ is also finite. a) The inequality above can be replaced by division if M5 L5" Ÿ K

(3.2)

for all 5 œ "ß á ß 8  ". b) Equality holds in Poincaré's inequality if and only if ÐM5 L5" À L5" Ñ œ ÐK À L5" Ñ for all 5 œ "ß á ß 8  ". Hence, if K is finite or if (3.2) holds for all 5 , then equality holds in Poincaré's inequality if and only if

Cosets, Index and Normal Subgroups

63

M5 L5" œ K for all 5 œ "ß á ß 8  ". 7) If a finite group K has subgroups L and O for which ÐK À LÑ and ÐK À OÑ are relatively prime, then K œ LO and equality holds in Poincaré's inequality, that is, ÐK À L ∩ OÑ œ ÐK À LÑÐK À OÑ Proof. We leave proof of part 1) and part 2) to the reader. For part 3), let M be a left transversal for KÎO and let N be a left transversal for OÎL . If + − K, then + œ 35 for some 5 − O and 5 œ 42 for some 4 − N , whence +L œ 35L œ 342L œ 34L Moreover, 34L œ 3w 4w L

Ê

3O ∩ 3w O Á g

3 œ 3w

Ê

and so 4L œ 4w L , whence 4 œ 4w . Thus, the set Ö34 ± 3 − Mß 4 − N × is a left transversal for KÎL . We leave proof of part 4) for the reader. For part 5), let M œ L ∩ O and consider the function 0 À LÎM Ä LOÎO defined by 0 Ð2MÑ œ 2O . This map is well defined and injective since if 2" ß 2# − L , then 2#" 2" − L implies that 2" M œ 2 # M

Í

2#" 2" − M

Í

2#" 2" − O

Í

2" O œ 2# O

Also, 0 is surjective since for any 2 − L and 5 − O , 0 Ð2MÑ œ 2O œ 25O Hence, ÐL À L ∩ OÑ œ ÐLO À OÑ. For part 6), we proceed by induction on 8. For 8 œ #, we have ÐK À L" ∩ L# Ñ œ ÐK À L" ÑÐL" À L" ∩ L# Ñ œ ÐK À L" ÑÐL" L# À L# Ñ Ÿ ÐK À L" ÑÐK À L# Ñ and the last inequality can be replaced by division if L" L# Ÿ K . Moreover, equality holds if and only if ÐL" L# À L# Ñ œ ÐK À L# Ñ. Assume the result is true for L" ß á ß L8" and let M8" œ L" ∩ â ∩ L8" . Then ÐK À M8" ∩ L8 Ñ œ ÐK À M8" ÑÐM8" À M8" ∩ L8 Ñ Ÿ ÐK À L" ÑâÐK À L8" ÑÐM8" À M8" ∩ L8 Ñ œ ÐK À L" ÑâÐK À L8" ÑÐM8" L8 À L8 Ñ Ÿ ÐK À L" ÑâÐK À L8 Ñ and both inequalities can be replaced with division signs if

64

Fundamentals of Group Theory

M5 L5" Ÿ K for all 5 œ "ß á ß 8  ". Moreover, equality holds if and only if ÐM5" L5 À L5 Ñ œ ÐK À L5 Ñ for all 5 . For part 7), since each of ÐK À LÑ and ÐK À OÑ divides ÐK À L ∩ OÑ and since these factors are relatively prime, we have ÐK À LÑÐK À OÑ ± ÐK À L ∩ OÑ Hence, Poincaré's inequality is an equality: ÐK À L ∩ OÑ œ ÐK À LÑÐK À OÑ and so the finiteness of K implies that K œ LO .… We have seen (Theorem 2.21) that any subgroup of a finitely-generated abelian group is finitely generated. We can now prove that if K is a finitely-generated group, then any subgroup of finite index is also finitely generated. Theorem 3.3 Let K be a finitely-generated group. If L is a subgroup of K of finite index, then L is also finitely generated. Proof. Let X œ Ö>" ß á ß >7 × be a left transversal for KÎL , with >" œ ". Let ÖB" ß á ß B8 × be a generating set for K and let " [ œ ÖB" ß á ß B8 × ∪ ÖB" " ß á ß B8 ×

Thus, if + − K, then + œ A: âA" for some A3 − [ . But A" œ >=" for some > − X and =" − L and so + œ A: âA# >=" We are now prompted to consider how the >'s and A's commute. For each > − X and A − [ , there exist unique >w − X and 2 − L for which A> œ >w 2 Let W © L be the finite set of all such 2's, as A varies over [ and > varies over X . Note that =" − W since A" >" œ A" œ >=" . By continually moving the element belonging to X forward in the product expression for +, we get + œ >w =: â=" − >w ØWÙ for some >w − X and =3 − W . But if + − L , then + − >w ØWÙ © >w L implies that >w œ ". Hence, L Ÿ ØWÙ and since W © L , we have L œ ØWÙ.…

Cosets, Index and Normal Subgroups

65

Quotient Groups and Normal Subgroups Let K be a group and let L Ÿ K. We have seen that the equivalence relation corresponding to the partition KÎL is equivalence modulo L : + ´ , mod L

if +L œ ,L

Now, there seems to be a natural way to “raise” the group operation from K to KÎL by defining (3.4)

+L ‡ ,L œ +,L

Of course, for this operation to make sense, it must be well defined, that is, we must have for all +ß +" ß ,ß ," − K , +"" + − L

and ,"" , − L

Ê

Ð+" ," Ñ" Ð+,Ñ − L

Taking +" œ " and ," œ , gives the necessary condition +−L

Ê

, " +, − L

(3.5)

However, this condition is also sufficient, since if it holds, then " " " " ,"" +" " +, œ Ð," +" ," ÑÐ," +," ÑÐ," ,Ñ − L

Note that (3.5) is equivalent to each of the following conditions: 1) +L+" © L for all + − K 2) +L+" œ L for all + − K 3) +L œ L+ for all + − K. Definition A subgroup L of K is normal in K, written L ü K, if +L œ L+ for all + − K. If L ü K and L Á K, we write L – K. The family of all normal subgroups of a group K is denoted by norÐKÑ.… Thus, the product (3.4) is well defined if and only if L ü K. Moreover, if L ü K, then for any +ß , − K, +,L œ +,LL œ +L,L that is, +L ‡ ,L œ +L,L In particular, the set product of two cosets of L is a coset of L . Moreover, if the set product of cosets is a coset, that is, if +L,L œ -L for some - − K, then +, − -L and so -L œ +,L , that is,

66

Fundamentals of Group Theory

+L,L œ +,L Let us refer to this as the coset product rule. Finally, if the coset product rule holds, then L ü K, since +L+" © +L+" L œ L for all + − K. Thus, the following are equivalent: 1) The binary operation +L ‡ ,L œ +,L is well defined on KÎL . 2) L ü K . 3) The set product of cosets is a coset. 4) The coset product rule holds. Moreover, if these conditions hold, then KÎL is actually a group under the set product, for it is easy to verify that the set product is associative, KÎL has identity element L and that the inverse of +L is +" L . Thus, we can add a fifth equivalent condition to the list above: 5) KÎL is a group under set product. Before summarizing, let us note that the following are equivalent: L +L , − +L , − +L + ´ , mod L

üK © L+ for all + − K Ê , − L+ for all +ß , − K Ê , " − +" L for all +ß , − K Ê +" ´ , " mod L for all +ß , − K

Also, the following are equivalent: The coset product rule holds +L,L © +,L for all +ß , − K w w + − +Lß , − ,L Ê +w , w − +,L for all +ß , − L w + ´ + mod Lß , w ´ , mod L Ê +w , w ´ +, mod L Now we can summarize. Theorem 3.6 Let L Ÿ K. The following are equivalent: 1) The set product on KÎL is a well-defined binary operation on KÎL . 2) The coset product rule +L,L œ +,L holds for all +ß , − L .

Cosets, Index and Normal Subgroups

67

3) L is a normal subgroup of K. 4) KÎL is a group under set product, called the quotient group or factor group of K by L . 5) The inverse preserves equivalence modulo L , that is, + ´ , mod L

+" ´ , " mod L

Ê

for all +ß , − K. 6) The product preserves equivalence modulo L , that is, +w ´ + mod Lß

, w ´ , mod L

Ê

+w , w ´ +, mod L

for all +ß +w ß ,ß , w − K.… When we use a phrase such as “the group KÎL ” it is the with the tacit understanding that L is normal in K. Note finally that statements 5) and 6) say that equivalence modulo L is a congruence relation on K. A congruence relation ) on an algebraic structure, such as a group, is an equivalence relation that preserves the (nonnullary) algebraic operations. Thus, a congruence relation ) on a group K must satisfy the conditions +) ,

+" ), "

Ê

and +) ,ß

-).

Ê

Ð+-Ñ)Ð,.Ñ

Theorem 3.6 shows that these two conditions are actually equivalent for groups.

More on Normal Subgroups There are several slight variations on the definition of normality that are often useful. We leave proof of the following to the reader. Theorem 3.7 Let L Ÿ K. The following are equivalent: 1) L ü K 2) L + © L for all + − K 3) L + ª L for all + − K 4) Every right coset of L is a left coset, that is, for all + − Kß there is a , − K such that L+ œ ,L 5) Every left coset is a right coset. 6) For all +ß , − K, +, − L

Ê

,+ − L

7) If + − K and 2 − L , then +2 œ 2 + for some 2 w − L .… w

Theorem 3.7 implies that a normal subgroup permutes with all subgroups of K. Hence, the normality of either factor guarantees that the set product LO is a subgroup.

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Fundamentals of Group Theory

Theorem 3.8 Let Lß O Ÿ K. 1) If either L or O is normal in K, then L and O permute and LO œ L ” O In this case, we refer to LO as the seminormal join of L and O . 2) If both L and O are normal in K, then LO is also normal in K and we refer to LO as the normal join of L and O . 3) The fact that LO ü K does not imply that either subgroup need be normal. Proof. For part 3), let K œ W% . Let L œ Ö+, Ð"#ÑÐ$%Ñß Ð"$ÑÐ#%Ñ, Ð"%ÑÐ#$Ñß Ð#%Ñß Ð"$Ñß Ð"#$%Ñß Ð"%$#Ñ× and O œ W$ œ Ö+ß Ð"#Ñß Ð"$Ñß Ð#$Ñß Ð"#$Ñß Ð"$#Ñ×

Then L ∩ O œ Ö+ß Ð"$Ñ× has size # and so kLO k œ Ð) ‚ 'ÑÎ# œ #% œ kW% k, which implies that LO œ W% . But neither subgroup is normal: L is not normal since Ð"%ÑÐ"$ÑÐ"%Ñ œ Ð%$Ñ Â L and O is not normal since Ð"%ÑÐ"#ÑÐ"%Ñ œ Ð%#Ñ Â W$ .… Example 3.9 (The normal subgroups of H#8 ) We have seen that for . ± 8, the subgroups of the dihedral group H#8 are 1) the cyclic subgroup Ø38Î. Ù of order . , 2) for each ! Ÿ 5  8Î. , the dihedral subgroup W œ 535 Ø38Î. Ù “ Ø38Î. Ù œ Ø535 ß 38Î. Ù of order #.. Subgroups of type 1) are normal, since conjugation gives 533 Ð3<8Î. Ñ33 5 œ 3<8Î. − Ø38Î. Ù Let W œ Ø535 ß 38Î. Ù be a subgroup of type 2). Then since Ø38Î. Ù ü W , it follows that W is normal if and only if the conjugates of the other generator 535 are in W . Conjugation by 3 gives 3Ð535 Ñ3" œ 53#5 which is in W if and only if 3#5 œ 3578Î. for some integer 7, that is, if and only if

Cosets, Index and Normal Subgroups

# ´ 7

69

8 mod 8 .

Multiplying both sides of this by . gives #. ´ ! mod 8, that is, 8 ± #. and since . ± 8, we must have 8 œ . or 8 œ #. . Conversely, if 8 œ . or 8 œ #. , then the congruence holds and W is closed under conjugation by 3. If 8 œ . , then W œ H#8 . If 8 œ #. , then 9ÐWÑ œ 8 and 5 œ ! or 5 œ ". If 5 œ !, then W œ Ø5ß 3# Ù and if 5 œ ", then W œ Ø53ß 3# Ù. Moreover, since ÐH#8 À WÑ œ #, both subgroups are normal, as we will prove in Theorem 3.17. Thus, the proper normal subgroups of H#8 are the subgroups of Ø3Ù and, for 8 even, the two subgroups Ø5ß 3# Ù and Ø53ß 3# Ù of order 8.…

Special Classes of Normal Subgroups There are two very important special classes of normal subgroups. Note that L ü K if and only if L is invariant under all inner automorphisms #+ of K. Definition Let K be a group. 1) A subgroup L of K is characteristic in K if it is invariant under all automorphisms of K. If L is characteristic in K, we write L « K. (This is not a standard notation, there being none.) We also write L ö K if L « K and L Á K. 2) A subgroup L of K is fully invariant in K if it is invariant under all endomorphisms of K.… Some of the most important subgroups of a group are characteristic. For example, the center ^ÐKÑ of a group K is characteristic in K .

The Lattice of Normal Subgroups of a Group If K is a group, then norÐKÑ is a subfamily of subÐKÑ and is partially ordered by set inclusion as well. Moreover, the intersection of any family Y of normal subgroups of K is normal in K and so the meet of Y in norÐKÑ is the same as the meet of Y in subÐKÑ. As to join, if Y œ ÖR3 ± 3 − M× is a nonempty family of normal subgroups of K, then the join of Y in the lattice subÐKÑ is the subgroup 2

3−M

R3 œ Ö+3" â+38 ± +35 − R35 ß 8   !×

Actually, Theorem 3.7 implies that we can collect factors from the same subgroup R3 and so 2

3−M

R3 œ Ö+3" â+38 ± +35 − R35 ß 35 Á 34 for 5 Á 4ß 8   !×

In particular, if Y œ ÖR" ß á ß R7 × is a finite family, then the join takes the particularly simple form

70

Fundamentals of Group Theory 2 Y œ Ö+" â+7 ± +3 − R3 ×

Now, if + − K, then

Š23−M R3 ‹ œ 23−M R3+ œ 23−M R3 +

and so the join of Y in subÐKÑ is normal in K. It follows that the join of Y in subÐKÑ is equal to the join of Y in norÐKÑ. Thus, ,

norÐKÑ

Y œ ,subÐKÑ Y

and

2

norÐKÑ

Y œ 2subÐKÑ Y

which holds also when Y is the empty family. Hence, norÐKÑ is a complete sublattice of subÐKÑ. Theorem 3.10 Let K be a group. 1) The subgroups Ö"× and K are normal in K. 2) If ÖR3 ± 3 − M× is a family of normal subgroups of K, then ,

3−M

R3

and

2

3−M

R3

are normal subgroups of K. Hence, norÐKÑ is a complete sublattice of subÐKÑ.… The maximal and minimal normal subgroups of a group play an important role in the theory. We state the definitions here for future use. Definition Let K be a group and let L ü K. 1) L is minimal normal if it is minimal in the partially ordered set of all nontrivial normal subgroups of K (under set inclusion). 2) L is maximal normal if it is maximal in the partially ordered set of all proper normal subgroups of K (under set inclusion).…

The Quasicyclic Groups For each prime :, we can now describe an infinite abelian group ™Ð:∞ Ñ, called the :-quasicyclic group that has a very interesting subgroup lattice. Specifically, the lattice of proper subgroups of ™Ð:∞ Ñ consists entirely of a single ascending chain of finite cyclic subgroups Ö"×  Ø+" Ù  Ø+# Ù  â We begin by looking at the quotient group  7 œ š ™ ¹ 7ß 8 − ™ß ! Ÿ 7  8ß Ð7ß 8Ñ œ "› ™ 8 The set

Cosets, Index and Normal Subgroups Wœš

71

7 ¹ 7ß 8 − ™ß ! Ÿ 7  8ß Ð7ß 8Ñ œ "› 8

is a left transversal for Ι and so we can simply identify Ι with W , under addition modulo ". Note that the order of 7Î8 − W is 8. Let : be a prime and let ™Ð:∞ Ñ be the subgroup of W consisting of those elements of order a power of :, that is, ™Ð:∞ Ñ œ œ

7 ¹ 7ß 5 − ™ß ! Ÿ 7  : 5 ß : y± 7 :5

If L  ™Ð:∞ Ñ and 7Î:5 − L for 7  !, then there are integers + and , for which +7  ,:5 œ " and so in ™Ð: ∞ Ñ, " +7  ,:5 +7 œ œ 5 −L 5 : :5 : Hence, ! Á 7Î:5 − L

Í

"Î:5 − L

Í

"Î:4 − L for all 4 Ÿ 5

Thus, since L is proper, there must be a largest integer 8 for which "Î:8 − L . Then "Î:5 − L

Í

5Ÿ8

Í

"Î:5 − Ø"Î:8 Ù

and so L œ Ø"Î:8 Ù is cyclic of order :8 . Hence, the proper subgroups of ™Ð:∞ Ñ are the subgroups Ö!×  Ø"Î:Ù  Ø"Î:# Ù  â In a later chapter, we will ask the reader to prove that the quasicyclic groups ™Ð:∞ Ñ are the only infinite groups (up to isomorphism) with the property that their proper subgroups consist entirely of a single ascending chain Ö!×  W"  W#  â

The Normal Closure of a Set If \ is a subset of a group K, then the smallest normal subgroup of K that contains \ is the intersection of all normal subgroups of K that contain \ . This subgroup is called the normal closure of \ in K and we will find it useful to use the following notations for this subgroup: \Kß

Ø\Ùnor

and

ncÐ\ß KÑ

The normal closure has a simple characterization as follows.

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Theorem 3.11 If \ is a nonempty subset of a group K, then the normal closure of \ is the subgroup ncÐ\ß KÑ œ ØB+ ± B − \ß + − KÙ generated by the conjugates of \ in K.… We can extend the notation and define for any \ß ] © K , \ ] œ ØBC ± B − \ß C − ] Ù This allows us to describe the join of two subgroups as a set product. Theorem 3.12 If Lß O Ÿ K, then ØLß OÙ œ L O O Proof. Since L O and O permute, it follows that L O O Ÿ K. Since L Ÿ L O O and O Ÿ L O O , it follows that ØLß OÙ Ÿ L O O . The reverse inclusion is clear.…

Internal Direct Products Strong Disjointness of a Family of Normal Subgroups If Y œ ÖL3 ± 3 − M× is a nonempty family of normal subgroups of a group K, we write LÐ3Ñ ³ 2ÖL4 ± 3 − Mß 4 Á 3×

for the join of all members of Y except L3 . The members of such a family Y can enjoy two levels of disjointness. The members of Y can be pairwise essentially disjoint, that is, L3 ∩ L4 œ Ö"× for all 3 Á 4. Note that Y is pairwise essentially disjoint if and only if 23 24 œ " for 23 − L3 and 24 − L4 with 3 Á 4 imply that 23 œ 24 œ ". Also, the members of a pairwise essentially disjoint family Y commute elementwise, that is, 23 24 œ 24 23 for all 23 − L3 and 24 − L4 , where 3 Á 4. A stronger level of disjointness comes when each L3 is essentially disjoint from the join of the other members of the family. The following useful definition is not standard in the literature. Definition We will say that a nonempty family Y œ ÖL3 ± 3 − M× of normal subgroups of a group K is strongly disjoint if

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L3 ∩ LÐ3Ñ œ Ö"× for all 3 − M .… The property of being strongly disjoint can be characterized as follows. Theorem 3.13 Let Y œ ÖL3 ± 3 − M× be a nonempty family of normal subgroups of a group K. Then the following are equivalent: 1) Y is strongly disjoint 2) If 23" â238 œ " where 234 − L34 and 34 Á 35 for 4 Á 5 , then 234 œ " for all 4. 3) Every nonidentity + − 1Y can be written, in a unique way except for the order of the factors, as a product + œ 23" â238 where " Á 234 − L34 and 34 Á 35 for 4 Á 5 . The element 234 is called the 34 th component of +. For 3 − M Ï Ö3" ß á ß 38 ×, the 3th component of + is ". Proof. If Y is strongly disjoint and 23" â238 œ ", where the factors are from different subgroups and 8   #, then 23" œ Ð23# â238 Ñ" − L3" ∩ LÐ3" Ñ œ Ö"× and so 23" œ ". Repeating this argument gives 234 œ " for all 4 and so 1) implies 2). If 2) holds, then the L3 's commute elementwise. To see that 3) holds, it is clear that every nonidentity + − 1Y has such a product representation. Moreover, if + has two such product representations, then we may include additional factors equal to " so that + œ 23" â238 œ 53" â538 where 234 ß 534 − L34 and 34 Á 35 for 4 Á 5 and at least one factor on each side is not equal to the identity. Then Ð53" 23" ÑâÐ53" 238 Ñ œ " " 8 and so 2) implies that 234 œ 534 for all 4. Hence 2) implies 3). Finally if 3) holds and + − L3 ∩ LÐ3Ñ for some 3, then the uniqueness condition implies that + œ ".…

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The next theorem says that strong disjointness is a finitary condition and that if Y is strongly disjoint, then it is relatively easy to check that Y “ ÖO× is also strongly disjoint. Theorem 3.14 Let Y œ ÖL3 ± 3 − M× be a nonempty family of normal subgroups of a group K. 1) Y is strongly disjoint if and only if every nonempty finite subset of Y is strongly disjoint. 2) Let O − norÐKÑ Ï Y . If Y is strongly disjoint, then Y “ ÖO× is strongly disjoint if and only if Š2Y ‹ ∩ O œ Ö"×

…

The following result can be quite useful. Theorem 3.15 Let Y œ ÖL3 ± 3 − M× be a nonempty family of normal subgroups of a group K. For any O ü K, there is a N © M that is maximal with respect to the property that the family YN œ ÖL4 ± 4 − N × ∪ ÖO× is strongly disjoint. Proof. Write \ œ ÖN © M ± YN is strongly disjoint× Then \ is nonempty and the union of any chain in \ is in \ . Hence, Zorn's lemma implies that \ has a maximal member.…

Internal Direct Products We have already discussed the external direct product K } L of two groups K and L . The internal direct product is defined as follows. Definition A group K is the (internal) direct product of two normal subgroups L and O if K œ L ì O . We use the notation K œ L  O to denote the internal direct product.… The internal direct product K œ L  O is a decomposition of K into an essentially disjoint product of normal subgroups. Since the factors L and O commute elementwise, the product in K takes the form Ð2" 5" ÑÐ2# 5# Ñ œ Ð2" 2# ÑÐ5" 5# Ñ where 23 − L and 53 − O . Thus, the groups L and O have the same level of independence as the factors in an external direct product. Indeed, the map 25 È Ð2ß 5Ñ is an isomorphism from L  O to L { O .

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Definition A nontrivial group K is said to be indecomposable if K cannot be written as an internal direct product of two proper subgroups, that is, KœL O

L œ K or O œ K

Ê

…

The internal direct product can easily be generalized to arbitrary nonempty families of normal subgroups. Definition A group K is the (internal) direct sum or (internal) direct product of a family Y œ ÖL3 ± 3 − M× of normal subgroups if Y is strongly disjoint and K œ 1Y . We denote the internal direct product of Y by  L3

or

Y

or when Y œ ÖL" ß á ß L8 × is a finite family, L"  â  L8 Each factor L3 is called a direct summand or direct factor of K. We denote the family of all direct summands of K ,C Wf ÐKÑ.… Theorem 3.13 implies the following. Theorem 3.16 Let Y œ ÖL3 ± 3 − M× be a nonempty family of normal subgroups of a group K. Then the following are equivalent: 1) K œ  Y 2) Every nonidentity + − K can be written, in a unique way except for the order of the factors, as a product + œ 23" â238 where " Á 234 − L34 and 34 Á 35 for 4 Á 5 .… A note on terminology is also in order. If Y œ ÖL3 ± 3 − M× is a family of normal subgroups of a group K, to say that the join 1L3 is direct in K or to say that the direct sum  L3 exists in K is the same as saying that Y is strongly disjoint and that the join is the direct sum.

Projection Maps Associated with an internal direct product K œ  ÖL3 ± 3 − M× is a family of projection maps. Specifically, the 3th projection map 33 À K Ä L3 is defined by setting 33 Ð+Ñ to be the 3th component of +. In this case, 33 can be thought of as an endomorphism of K and then the following hold: 1) Ð33 ÑlL3 œ +L3

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2) 33 34 œ ! if 3 Á 4, where ! is the zero map 3) 33 is idempotent, that is, 33# œ 33 . Note also that if 3 Á 4, then the images L3 œ imÐ33 Ñ and L4 œ imÐ34 Ñ commute elementwise. We will have much more to say about the direct product in a later chapter.

Chain Conditions and Subnormality Sequences of Subgroups and the Chain Conditions Ordered sequences of subgroups play a key role in group theory. For infinite ascending and descending sequences of subgroups, the issue centers around the chain conditions. Generally speaking, chain conditions are considered a form of finiteness condition on a group and we will study the consequences of the chain conditions on various families of subgroups, such as the family of all subgroups, all normal subgroups or all subnormal subgroups throughout the book. For the record, here is the definition, which will be repeated later. Definition Let K be a group and let f be a family of subgroups of K. 1) A group K satisfies the ascending chain condition (ACC) on f if every ascending sequence L" Ÿ L# Ÿ â of subgroups in f must eventually be constant, that is, if there is an 8  ! such that L85 œ L8 for all 5   !. In this case, we also say that f has the ACC. 2) A group K satisfies the descending chain condition (DCC) on f if every descending sequence L"   L#   â of subgroups in f must eventually be constant, that is, if there is an 8  ! such that L85 œ L8 for all 5   !. In this case, we also say that f has the DCC. 3) A group K satisfies both chain conditions (BCC) on f if K has the ACC and the DCC on f . In this case, we also say that f has BCC.…

Finite Series and Subnormality Finite ordered sequences of subgroups of a group are just as important as infinite sequences, but rather than conveying any finiteness condition about a group, they convey structural information about the group and are used to classify groups via this structure. For example, a group K that has a finite sequence Ö"× œ L! ü L" ü â ü L8 œ K

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of subgroups for which each quotient L5" ÎL5 is abelian is called a solvable group. Solvable groups play a key role in the Galois theory of fields and we will study them in detail later in the book. Unfortunately, the terminology surrounding finite ordered sequences of subgroups is not at all standardized. For example, consider the following types of finite sequences of subgroups of a group K: 1) An arbitrary nondecreasing sequence of subgroups of K: K! Ÿ K " Ÿ â Ÿ K 8 2) A sequence of subgroups of K of the form K! ü K" ü â ü K 8 in which each subgroup is normal in its immediate successor. 3) A sequence of subgroups of K of the form K! Ÿ K " Ÿ â Ÿ K 8 in which each subgroup is normal in the parent group K. Some authors refer to 1) as a series, 2) as a subnormal series and 3) as a normal series. Some authors refer to 2) as a series and 3) as a normal series. Some authors refer to 2) as a normal series and 3) as an invariant series. Since arbitrary finite nondecreasing sequences of subgroups are a bit too general to be really useful, it seems reasonable not to give them a special name and simply refer to them as sequences, thus reserving the term series for more useful types of sequences. Accordingly, we choose the following terminology. Definition Let K be a group and let K! ß K8 Ÿ K. A series in K from K! to K8 is a sequence of subgroups of K of the form K! ü K" ü â ü K 8 where K5 is normal in K5" for each 5 . Each group K5 is a term in the series; K! is the lower endpoint of the series and K8 is the upper endpoint. Each extension K5 ü K5" is a step in the series. A series is proper if each inclusion is proper. The length of a series is the number of proper inclusions. 1) A normal series in K is a series in K in which each term is normal in the parent group K. 2) A characteristic series in K is a series in K in which each term is characteristic in the parent group K. 3) A fully-invariant series in K is a series in K in which each term is fully invariant in the parent group K.…

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The following generalization of normality is extremely important and we shall have much to say about it in this book. Definition A subgroup L Ÿ K is subnormal in K, written L ü ü K, if there is a series L ü L" ü â ü L8 œ K from L to K. If L ü ü K and L  K, we write L – – K. The family of subnormal subgroups of K is denoted by subnÐKÑ.… Thus, L ü ü K if there is a sequence of “normal steps” from L to K.

Subgroups of Index # The largest proper subgroups of a group are the subgroups of index #. We can now improve upon Theorem 2.28. Theorem 3.17 Let L be a subgroup of K of index #. 1) L ü K . 2) If + − K, then +# − L . 3) If K is finite, then any subgroup W of K is either a subgroup of L or else kW ∩ L k œ kW kÎ#

In words, W lies completely in L or else W lies half-in and half-out of L . Also, if + − W Ï L , then W œ ÐW ∩ LÑ “ +ÐW ∩ LÑ where “ is the disjoint union. Proof. For part 1), if +  L , then ÖLß +L× and ÖLß L+× are both partitions of K and so +L œ L+ for all + − K . Hence, L ü K . For part 2), if +  L then +L Á L has order # in KÎL and so +# L œ Ð+LÑ# œ L , which implies that +# − L . For part 3), if W is not contained in L , the normality of L implies that WL œ K and so ÐW À L ∩ WÑ œ ÐLW À LÑ œ ÐK À LÑ œ #

…

Example 3.18 For 8   #, the alternating group E8 has index # in the symmetric group W8 and so E8 ü W8 .… We can now show that the converse of Lagrange's theorem fails. Example 3.19 The alternating group E% has order %xÎ# œ "#, but has no subgroups of order '. For if L Ÿ E% has order ', then L has index # and so 5# − L for all 5 − E% . But there are eight $-cycles 5 in E% and each one is a square:

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5 œ 5 % œ Ð5 # Ñ# − L Since these ) elements cannot fit into a subgroup of size ', there can be no such subgroup.…

Cauchy's Theorem We have seen that the converse of Lagrange's theorem fails to hold. However, there are some partial converses to Lagrange's theorem. For example, there are certain classes of groups for which the converse of Lagrange's theorem does hold. In particular, we will see later in the book that if K is a finite abelian group or if 9ÐKÑ œ :8 where : is prime, then the converse of Lagrange's theorem holds, that is, K has a subgroup of any order 5 that divides 9ÐKÑ. On the other hand, it is true that if a prime : divides 9ÐKÑ, then K has an element of order : and hence a subgroup of order :. This key theorem is called Cauchy's theorem. Cauchy's theorem has a very colorful history, beginning with its discovery by Cauchy in 1845 as the main conclusion of a 101-page paper ([6]). Here is a quotation from the abstract of an article on the history of Cauchy's theorem by M. Meo [24]: The initial proof by Cauchy, however, was unprecedented in its complex computations involving permutational group theory and contained an egregious error. A direct inspiration to Sylow’s theorem, Cauchy’s theorem was reworked by R. Dedekind, G. F. Frobenius, C. Jordan, and J. H. McKay in ever more natural, concise terms. The proof we give below is essentially the proof of J. H. McKay [23], which first appeared in 1959 and reminds us that we should never stop looking for “better” proofs of even the most basic results. Theorem 3.20 (Cauchy's theorem) Let K be a finite group. If : is a prime dividing 9ÐKÑ, then K has an element of order :. Proof. The key to the proof is to examine the set \ œ ÖÐ+" ß á ß +: Ñ ± +3 − Kß +" â+: œ "× which is nonempty, since Ð"ß á ß "Ñ − \ . In fact, the size of \ is easily computed by observing that the first :  " coordinates any B − \ can be assigned arbitrarily and this uniquely determines the final coordinate. Hence, k\ k œ kKk:"

which is divisible by :.

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Now, if we can find a constant :-tuple Ð+ß á ß +Ñ in \ where + Á ", then +: œ " and so 9Ð+Ñ œ :, which proves the theorem. Note that a :-tuple B œ Ð+" ß á ß +: Ñ is constant if and only if rotation one position to the right (with wrap around) has no effect, that is, if and only if Ð+: ß +" ß á ß +:" Ñ œ Ð+" ß á ß +: Ñ We can put this in group-theoretic language as follows. Let each 5 − W: act on \ by permuting the coordinates. Thus, if 5 is the :-cycle Ð" # â :Ñ, then 5Ð+" ß á ß +: Ñ œ Ð+: ß +" ß á ß +:" Ñ and so B is constant if and only if 5B œ B, that is, if and only if 5 fixes B. Let us consider how the powers of 5 act on an element B − \ . The :-tuple 5 5 B comes from B by a 5 -fold rotation. The orbit of B − \ is the collection b ÐBÑ œ ÖBß 5Bß á ß 5 :" B× of the various rotated versions of B. Now, the primeness of : implies that the elements of b ÐBÑ are either all distinct or all the same. For if 53 B œ 54 B for 3  4, then 534 B œ B where !  3  4  :. If 5 œ 3  4, then Ð5ß :Ñ œ " implies that ?5  @: œ " for some ?ß @ − ™ and so 5B œ 5?5@: B œ 5?5 5?: B œ Ð55 Ñ? B œ B whence 53 B œ B for all 3. Thus, bÐBÑ has size " or : for all B − \ . Now, the orbits bÐBÑ are the equivalence classes of the equivalence relation on \ defined by B´C

if C œ 55 B for some 5 − ™

and so the distinct orbits form a partition of \ . If , is the number of blocks of size " and - is the number of blocks of size :, then k\ k œ ,  -:

where ,   ", since ÖÐ"ß á ß "Ñ× is a block of size ". Hence, : divides k\ k implies that : ± , and so, in particular, ,   :, which implies that there are at least :  " constant :-tuples Ð+ß á ß +Ñ with + Á ".…

Application to :-Groups The following types of groups play a key role in the study of the structure of finite groups. Definition Let K be a nontrivial group and let : be a prime. 1) An element + − K is called a :-element if 9Ð+Ñ œ :5 for some 5   !.

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2) K is a :-group if every element of K is a :-element. 3) A nontrivial subgroup W of K is called a :-subgroup of K if W is a :group.… Lagrange's theorem and Cauchy's theorem combine to describe finite :-groups quite succinctly. Theorem 3.21 A finite group K is a :-group if and only if the order of K is a power of :. Proof. If 9ÐKÑ œ :5 , then Lagrange's theorem implies that every element of K is a :-element and so K is a :-group. Conversely, if K is a finite :-group but ; ± 9ÐKÑ where ; Á : is prime, then Cauchy's theorem implies that K has an element of order ; , which is false. Hence, 9ÐKÑ œ :5 for some 5   ".…

The Center of a Group; Centralizers We briefly mentioned the following concept earlier. Definition The center ^ÐKÑ of a group K is the set of all elements of K that commute with all elements of K, that is, ^ÐKÑ œ Ö+ − K ± +, œ ,+ for all , − K× A group K is centerless if ^ÐKÑ œ Ö"×. A subgroup L of K is central if L is contained in the center of K.… It is easy to see that the center of K is a normal subgroup of K. Example 3.22 a) The center of the quaternion group U is ^ÐUÑ œ Ö"ß "×. b) For 8   $ odd, the dihedral group H#8 is centerless and for 8   $ even, ^ÐH#8 Ñ œ Ö+ß 38Î# × c)

For 8   $, the symmetric group W8 is rather large and it should come as no surprise that W8 is centerless. To see this, suppose that 5 − W8 is not the identity. If the cycle decomposition of 5 contains a 5 -cycle with 5   $, that is, if 5 œ âÐ+ , - âÑâ then 5Ð+ ,Ñ Á 5 and so Ð+ ,Ñ5 Á 5Ð+ ,Ñ. On the other hand, if the cycle decomposition of 5 is a product of disjoint transpositions: 5 œ Ð+ ,Ñâ then 5Ð, -Ñ Á 5 and so Ð, -Ñ5 Á 5Ð, -Ñ for -  Ö+ß ,×. In either case, 5  ^ÐW8 Ñ and so W8 is centerless.

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d) We will prove in a later chapter that for 8 Á %, the alternating group E8 is simple, that is, E8 has no nontrivial proper normal subgroups. Hence, E8 is centerless for 8   %.…

The Number of Conjugates of an Element Definition Let K be a group. The centralizer of an element , − K is the set of all elements of K that commute with , : GK Ð,Ñ œ Ö+ − K ± +, œ ,+× The centralizer of a subgroup L Ÿ K is the set

GK ÐLÑ œ ,+−L GK Ð+Ñ

of all elements of K that commute with every element of L .… It is easy to see that centralizers are subgroups of the parent group. Let K be a group and let + − K. Then +B œ + C Í + C

"

B

œ + Í C" B − GK Ð+Ñ Í BGK Ð+Ñ œ CGK Ð+Ñ

and so the element + has precisely ÐK À GK Ð+ÑÑ distinct conjugates. This formula is of considerable importance in the study of finite groups. Theorem 3.23 Let K be a group and let + − K. Let conjK Ð+Ñ denote the set of conjugates of + in K. Then kconjK Ð+Ñk œ ÐK À GK Ð+ÑÑ

which divides 9ÐKÑ when K is finite. The set conjK Ð+Ñ is called a conjugacy class in K.…

The Normalizer of a Subgroup Suppose that L Ÿ K. Then of course L is normal in itself. But it may also be normal in a larger subgroup of K. Definition Let K be a group and let L Ÿ K. The largest subgroup RK ÐLÑ of K for which L ü RK ÐLÑ is called the normalizer of L in K. A subset \ © K is said to normalize L if \ © RK ÐLÑ.… Theorem 3.24 Let K be a group. The normalizer of L Ÿ K is RK ÐLÑ œ Ö+ − K ± L + œ L×

…

Will see in the chapter on free groups (Theorem 12.21) that it is possible to have L + § L , in which case + Â RK ÐLÑ. However, since

Cosets, Index and Normal Subgroups

L+ œ L

Í

L+ © L

83

"

and L + © L

if a subset W © K is closed under inverses and has the property that L = © L for all = − W , then W © RK ÐLÑ. In particular, since W œ Ö+ − K ± L + © L× is closed under products, if W is finite then W is a subgroup of K and so W is closed under inverses, whence W œ RK ÐLÑ. In particular, if K is finite, then W œ RK ÐLÑ. The normalizer of L should not be confused with the normal closure of L , which is the smallest normal subgroup of K that contains L . Note that if O Ÿ K normalizes L Ÿ K , then LO is also a subgroup of K. Theorem 3.25 If K is a group and L Ÿ K, then GK ÐLÑ ü RK ÐLÑ

…

Elementwise Commutativity It is clear that L and O commute elementwise if and only if L Ÿ GK ÐOÑ and

O Ÿ GK ÐLÑ

For essentially disjoint subgroups, we need only check that each subgroup is contained in the normalizer of the other. Proof of the following is left to the reader. Theorem 3.26 Let L and O be essentially disjoint subgroups of a group K. 1) Then L and O commute elementwise if and only if each subgroup is contained in the normalizer of the other, that is, L Ÿ RK ÐOÑ and O Ÿ RK ÐLÑ In particular, if Lß O ü K, then L and O commute elementwise. 2) If K œ L ì O , then L and O commute elementwise if and only if Lß O ü K.…

The Number of Conjugates of a Subgroup Let K be a group and let L Ÿ K. Then LB œ LC Í LC

"

B

œ L Í C" B − RK ÐLÑ Í BRK ÐLÑ œ CRK ÐLÑ

Thus, we obtain a count of the number of conjugates of a subgroup (the analog of Theorem 3.23). Theorem 3.27 Let K be a group and let L Ÿ K . Let conjK ÐLÑ denote the set of conjugates of L in K. Then

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Fundamentals of Group Theory kconjK ÐLÑk œ ÐK À RK ÐLÑÑ

which divides 9ÐKÑ when K is finite. The set conjK ÐLÑ is called a conjugacy class in subÐKÑ.…

Simple Groups Nontrivial groups with no nontrivial proper normal subgroups are, in some sense, simple. Definition A nontrivial group K is simple if it has no normal subgroups other than Ö"× and K.… Of course, if K is an abelian group, then all of its subgroups are normal and so an abelian group is simple if and only if its only subgroups are Ö"× and K. This is not easy for a group. Theorem 3.28 An abelian group K is simple if and only if it is a cyclic group of prime order. Proof. If K is cyclic of prime order, then Lagrange's theorem implies that K has no subgroups of order different from " and 9ÐKÑ and so K is simple. Conversely, if K is simple, then every nonidentity element 1 of K must generate K, that is, K œ Ø1Ù is cyclic. However, if K is infinite, then Ø1# Ù is a nontrivial proper subgroup of K, a contradiction. Hence, K is finite and cyclic. But if : ± 9ÐKÑ where : is prime, then Cauchy's theorem implies that K has a subgroup W of order : and so K œ W has prime order.… We will show later in the book that the alternating group E8 is simple for all 8 Á %. Also, a famous result of Feit–Thompson (1963, [11]), whose proof runs 255 pages, says that every nonabelian finite simple group has even order. Thus: 1) The abelian simple groups are the cyclic groups of prime order. 2) All nonabelian finite simple groups have even order.

Commutators of Elements The elements of a group of the form +,+" , " play a special role. Definition Let K be a group. The commutator of +ß , − K is the element Ò+ß ,Ó œ +,+" , " We denote the set of commutators of K by VÐKÑ. The subgroup generated by all commutators of K is usually denoted (unfortunately) by Kw and is called the commutator subgroup, or derived subgroup of K. A group K is perfect if K œ Kw .…

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We should note that authors who define the conjugate of two elements by +, œ , " +, also define the commutator by Ò+ß ,Ó œ +" , " +, . It is easy to see that Kw is fully invariant in K ; in particular, Kw « K . Also, we leave it as an exercise to prove that if R ü K, then Œ

K R

w

œ

Kw R R

Commutators have some very nice algebraic properties. Here are the most basic of these properties. Theorem 3.29 Let K be a group and let +ß ,ß - − K. 1) Ò+ß ,Ó œ " Í +, œ ,+ 2) Ò+ß ,Ó" œ Ò,ß +Ó 3) Ò+ß ,Ó- œ Ò+- ß , - Ó 4) If L ü K, then in KÎL , Ò+Lß ,LÓ œ Ò+ß ,ÓL 5) If +ß , − K commute with -ß . − K and vice versa, then …

Ò+ß ,ÓÒ-ß .Ó œ Ò-+ß .,Ó

Note that since Ò+ß ,Ó" œ Ò,ß +Ó, every element of Kw is a product of commutators. The following characterization of commutator subgroups explains why these subgroups are so important. Theorem 3.30 (R. Dedekind, c. 1880) Let K be a group. Then for any subgroup L Ÿ K, L ü K and KÎL is abelian

Í

Kw Ÿ L

In particular, Kw is the smallest normal subgroup of K whose quotient is abelian. Proof. If L ü K and KÎL is abelian, then in KÎL , we have Ò+ß ,ÓL œ Ò+Lß ,LÓ œ L and so Ò+ß ,Ó − L , whence Kw Ÿ L . Conversely, if Kw Ÿ L , then L is normal in K, since for any 2 − L and + − K, 2+ œ Ò+ß 2Ó2 − L Also, KÎL is abelian since Ò+Lß ,LÓ œ Ò+ß ,ÓL œ L

…

Example 3.31 We leave it to the reader to show that the commutator subgroup of the quaternion group U is Uw œ Ö"ß "×, which is also the set of

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commutators, that is, Uw œ VÐUÑ w Similarly, the commutator subgroup of the dihedral group H#8 is H#8 œ Ø3# Ù and so again the commutator subgroup of H#8 is also the set of commutators:

Hw#8 œ VÐH#8 Ñ

…

Example 3.32 For the symmetric group W8 on 8   $ symbols, we have W8w œ E8 œ VÐW8 Ñ Since all commutators are even permutations, we have W8w Ÿ E8 . Now we make a few observations. First the product of any finite number of disjoint commutators is a commutator, since if α3 "3 and α4 "4 are disjoint, then Òα3 ß "3 ÓÒα4 ß "4 Ó œ Òα4 α3 ß "4 "3 Ó Second, if 5ß . − W8 , then 5. 5" œ Ò.ß 5Ó and so if 5 and 7 have the same cycle structure, then the product 75 is a commutator. Hence, we need only show that any 5 − E8 can be written as a product of disjoint permutations, each of which is a product α3 "3 where α3 and "3 have the same cycle structure. But since a cycle of odd length is even and a cycle of even length is odd, the cycle decomposition of 5 must have an even number of cycles of even length. In other words, the cycle decomposition of 5 is a product of odd cycles and pairs of even cycles. Now, every odd cycle 5 œ Ð+" â+#7" Ñ can be written as a product of two equal-length cycles as follows: 5 œ Ð+" â+#7" Ñ œ Ð+" â+7" ÑÐ+7" â+#7" Ñ Similarly, every pair of disjoint even cycles can be written as a product of two equal-length cycles as follows (where 5   7): 5 œ Ð+" â+#7 ÑÐ," â,#5 Ñ œ Ð+" â+#7 ," â,57" ÑÐ+#7 ,57" â,#5 Ñ Hence, 5 can be written in the form 5 œ Ðα" "" ÑâÐα7 "7 Ñ

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where α3 and "3 are equal-length cycles for each 3 and α3 "3 and α4 "4 are disjoint for 3 Á 4. Hence, 5 − VÐW8 Ñ.…

When is VÐKÑ œ Kw ? We have seen that for the quaternion, dihedral and symmetric groups, Kw œ VÐKÑ Despite these examples, however, this is not always true. In fact, the question of precisely when Kw œ VÐKÑ is an active area of current research. We give an example of a group for which Kw Á VÐKÑ and then discuss a few results in this area. Readers interested in more details may wish to consult the survey article of Kappe and Morse [19]. Example 3.33 (Cassidy [5]) Let K be the set of all matrices of the form 7Ð0 ß 1ß 2Ñ ³

Ô" ! Õ!

0 ÐBÑ " !

2ÐBß CÑ × 1ÐCÑ " Ø

where 0 ÐBÑ, 1ÐBÑ and 2ÐBß CÑ are polynomials with rational coefficients. A straightforward calculation shows that 7Ð0" ß 1" ß 2" Ñ7Ð0# ß 1# ß 2# Ñ œ 7Ð0"  0#ß 1"  1#ß 2"  2#  0 "1 #Ñ and 7Ð0 ß 1ß 2Ñ" œ 7Ð0 ß 1ß 2  0 1Ñ from which it follows that K is a group. Another computation shows that the commutators are given by Ò7Ð0" ß 1" ß 2" Ñß 7Ð0# ß 1# ß 2# ÑÓ œ 7Ð!ß !ß 0" 1#  0# 1" Ñ Thus, the commutator subgroup Kw is contained in the subgroup of all matrices in K of the form 7Ð!ß !ß 2Ñ where 2 − ÒBß CÓ. Moreover, any such matrix is a product of commutators, since +3ß4 B3 C4

7 !ß !ß 3ß4

œ $ 7Ð+3ß4 B3 ß !ß !Ñß 7Ð!ß C4 ß !Ñ‘ 3ß4

which shows that Kw œ Ö7Ð!ß !ß 2Ñ ± 2 − ÒBß CÓ× Thus, the matrix E œ 7Ð!ß !ß "  BC  B# C# Ñ is in Kw . However, it is not a commutator, for if

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"  BC  B# C# œ 0" ÐBÑ1# ÐCÑ  0# ÐBÑ1" ÐCÑ for some 0" ß 0# − ÒBÓ and 1" ß 1# − ÒCÓ, then equating coefficients of B3 shows that C3 œ 0"ß3 1# ÐCÑ  0#ß3 1" ÐCÑ for 3 œ !ß " and #, where 0?ß3 is the coefficient of B3 in 0? ÐBÑ. But this implies that the two-dimensional vector subspace of ÒCÓ spanned by 1" ÐCÑ and 1# ÐCÑ contains the three independent vectors ", C and C# , which is not possible. Hence, not all members of Kw are commutators.… Here is a small sampling of known results concerning the issue of when Kw œ VÐKÑ. Many more results are contained in Kappe and Morse [19]. Theorem 3.34 (Speigel [31], 1976) If a group K contains a normal abelian subgroup E whose quotient KÎE is cyclic, then Kw œ VÐKÑ. Proof. If we can find a normal subgroup F ü K for which F © VÐKÑ and KÎF is abelian, then Kw Ÿ F © VÐKÑ © Kw whence Kw œ VÐKÑ. To this end, let KÎE œ ØBEÙ and let F œ ÖÒBß +Ó ± + − E× œ Ö+B +" ± + − E× © VÐKÑ To see that F ü K , we have Ð+B +" ÑÐ, B , " Ñ œ +B , B +" , " œ Ð+,ÑB Ð+,Ñ" − F and Ð+B +" Ñ" œ Ð+" ÑB + − F and for normality, 3

3

3

3

Ð+B +" ÑB + œ Ð+B +" ÑB œ Ð+B ÑB Ð+B Ñ" − F To see that KÎF is abelian, we have ÒBFß +FÓ œ ÒBß +ÓF œ F and so BF and +F commute for all + − E, which implies that B3 +F and B4 ,F commute for all +ß , − E.… For small groups, we also have Kw œ VÐKÑ. Theorem 3.35 Let K be a group. The following conditions imply that Kw œ VÐKÑ. 1) (Guralnick [15], 1980)

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a) Kw is abelian and either 9ÐKÑ  "#) or 9ÐKw Ñ  "'. b) Kw is nonabelian and either 9ÐKÑ  *' or 9ÐK wÑ  #% . 2) (Kappe and Morse [20], 2005) a) 9ÐKÑ œ :8 , where : is an odd prime and 8 Ÿ &. b) 9ÐKÑ œ #8 , where 8 Ÿ '.… On the other hand, if the center of K is large compared to the size of Kw , then there will be elements of Kw that are not commutators. Theorem 3.36 (MacDonald [22], 1986) Let K be a group with ^ œ ^ÐKÑ. If ÐKÀ ^Ñ#  9ÐKw Ñ then Kw Á VÐKÑ. Proof. Since Dß A − ^ implies that Ò+Dß ,AÓ œ Ò+ß ,Ó it follows that the number of distinct commutators is at most the number of commutators of KÎ^ and that is certainly at most ÐKÀ ^Ñ# .…

Additional Properties of Commutators Here are some additional properties of commutators. The reader may wish simply to read the statement of the theorem and move on, referring to the theorem as needed at later times. (The proof is not particularly enlightening.) Theorem 3.37 Let K be a group and let +ß ,ß - − K. 1) Ò,ß +Ó œ Ò+" ß ,Ó+ œ Ò+ß , " Ó, Ò+ß ,Ó œ Ò+" ß , " Ó,+ 2) Ò+ß ,-Ó œ Ò+ß ,ÓÒ+ß -Ó, Ò+,ß -Ó œ Ò,ß -Ó+ Ò+ß -Ó 3) Let +" ß á ß +8 ß ," ß á ß ,7 − K . Then Ò+" â+8 ß ," â,7 Ó œ $$ Ò+3 ß ,4 Óα3ß4 8

7

3œ" 4œ"

where α3ß4 − Ø+" ß á ß +8 ß ," ß á ß ,7 Ù. Also, the order of the factors on the right can be chosen arbitrarily, although the exponents depend on that order. In particular, if 7 and 8 are positive integers, then

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Ò+8 ß , 7 Ó œ $$ Ò+ß ,Óα3ß4 8

7

3œ" 4œ"

where α3ß4 − Ø+ß ,Ù. 4) If Ò+ß ,Ó commutes with both + and , , then for any integers 7 and 8, a) Ò+7 ß , 8 Ó œ Ò+ß ,Ó78 7 b) Ð+,Ñ7 œ +7 , 7 Ò,ß +Ó # Hence, if 9ÐÒ+ß ,ÓÑ divides ˆ 7# ‰, then Ð+,Ñ7 œ +7 , 7 Proof. The proofs of part 1) and part 2) are straightforward calculations. Part 3) is proved by a double induction. For 7 œ ", we induct on 8. If 8 œ ", then the result is clear. If the result holds for 8  ", then part 2) implies that Ò+" â+8 ß ," Ó œ Ò+8 ß ," Ó+" â+8" Ò+" â+8" ß ," Ó and the inductive hypothesis completes the proof for 7 œ ". Assume the result is true for 7  " and let + œ +" â+8 . Then Ò+ß ," â,7 Ó œ Ò+ß ," â,7" ÓÒ+ß ,8 Ó," â,7" and the inductive hypothesis completes the proof. As to the statement about the order, we can rearrange the factors using the identity BC œ CB B. Part 4a) follows from part 3) when 7ß 8   !. The result then follows for negative exponents using part 1). For part 4b), note first that - œ Ò,ß +Ó also commutes with + and , and that ,+ œ -+, œ +,Now,

Ð+,Ñ7 œ ðóóóóóñóóóóóò Ð+,ÑâÐ+,ÑÐ+,Ñ 7 factors

and if we move the rightmost + all the way to the left, the result is Ð+,Ñ7 œ +ðóóñóóò Ð+,ÑâÐ+,Ñ ,- 7" 7" factors

Repeating this process gives

Ð+,Ñ7 œ +# ðóóñóóò Ð+,ÑâÐ+,Ñ , 2 - 7# - 7" 7# factors

Continuing until all of the +s are on the left gives Ð+,Ñ7 œ +7" Ð+,Ñ, 7" - "#âÐ7"Ñ œ +7 , 7 -

7 #

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which proves the result for 7   !. For 7  ! we have, using part 4b) for positive exponents along with part 4a) and part 1), Ð+,Ñ7 œ Ð, " +" Ñ7 7 œ , 7 +7 Ò+" ß , " Ó # œ +7 , 7 Ò, 7 ß +7 ÓÒ+" ß , " Ó #

œ +7 , 7 Ò,ß +Ó7 Ò+" ß , " Ó #

œ +7 , 7 Ò,ß +Ó7 Ò+ß ,Ó #

7 #

7 #

7 #

œ +7 , 7 Ò,ß +Ó7 Ò,ß +Ó 7 œ +7 , 7 Ò,ß +Ó #

7 #

…

Here is one application of Theorem 3.37 that will be useful later in the book. Theorem 3.38 If K is a nonabelian group with the property that all cyclic subgroups are normal, then 1) K is periodic 2) Any nonabelian subgroup of K contains a quaternion subgroup 3) If +ß , − K and 9Ð+Ñ  9Ð,Ñ  ), then +, œ ,+. Proof. To see that K is periodic, we first show that any B  ^ÐKÑ has finite order. Let C − K satisfy - ³ ÒBß CÓ Á " Then the normality of ØBÙ and ØCÙ imply that - − ØBÙ ∩ ØCÙ and so - œ B8 œ C 7 for some 8ß 7 Á !. Thus, - commutes with B and C and so " œ Ò-ß -Ó œ ÒB8 ß C7 Ó œ ÒBß CÓ87 œ - 87 Hence, - has finite order and therefore so does B. Now let B − ^ÐKÑ . If C  ^ÐKÑ, then BC  ^ÐKÑ and so BC has finite order and therefore so does B. Hence, K is periodic. Now suppose that H is a nonabelian subgroup of K. Let Bß C − H be chosen so that 9ÐBÑ  9ÐCÑ is minimal among all pairs of noncommuting elements in H. We will show that U œ ØBß CÙ is a quaternion subgroup of H. To see that B and C have order %, let : be a prime dividing 9ÐBÑ and let - œ ÒBß CÓ œ B8 œ C7 as shown above. Since 9ÐB: Ñ  9ÐBÑ, it follows that B: and C commute and so Theorem 3.37 implies that " œ ÒB: ß CÓ œ ÒBß CÓ: whence 9Ð-Ñ œ : . Hence, : is the only prime dividing 9ÐBÑ and similarly 9ÐCÑ

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and so 9ÐBÑ œ :3"

and 9ÐCÑ œ :4"

for some 3ß 4   !. Moreover, :3" Ð:3" ß 8Ñ

: œ 9Ð-Ñ œ 9ÐB8 Ñ œ

which shows that 8 œ :3 ? where : y± ?. A similar argument for C gives 3

4

B: ? œ - œ C : @ where : y± ? and : y± @. To eliminate ? and @ from the equation above, if α œ ?" mod : and " œ @" mod :, then taking the α" power gives 3

4

B: " œ - α" œ C: α and so 3

ÐB" Ñ: œ ÒB" ß Cα Ó œ ÐCα Ñ:

4

where 9ÐB" Ñ œ :3" and 9ÐCα Ñ œ :4" . Thus, replacing B" by B and C α by C gives 3

B: œ - œ C:

4

(3.39)

where " Á - œ ÒBß CÓ and 9ÐBÑ œ :3" ß

9ÐCÑ œ :4"

and 9Ð-Ñ œ :

We may also assume that 3   4   ", for if 4 œ !, then C œ - œ ÒBß CÓ, which is false. The next step is to show that : œ # and that 3 œ 4 œ ". The key observation here is that for any integer 5 , the elements B5 C and B do not commute and so 9ÐB5 CÑ   9ÐCÑ œ :4" . But, if : is odd or if : œ # and 4  ", then 9Ð-Ñ œ : ¹ Œ

:4 #

and so Theorem 3.37 implies that 4

4

4

4

" Á ÐB5 CÑ: œ B5: C: œ B5: and taking 5 œ :34 gives " Á ". Hence, : œ # and 3 œ 4 œ ". Thus, 9ÐBÑ œ 9ÐCÑ œ %ß

9Ð-Ñ œ #

and B# œ - œ C#

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where - commutes with B and C and since BCÐBCÑ œ -CBÐBCÑ œ it follows that U œ ØBß CÙ is quaternion.…

Commutators of Subgroups Commutators can be defined for subsets as well as for elements. Definition The commutator Ò\ß ] Ó of two subsets \ß ] © K is the subgroup generated by the commutators: Ò\ß ] Ó œ ØÒBß CÓ ± B − \ß C − ] Ù

…

Note that, by definition, Kw œ ÒKß KÓ for any group K. Here are some of the basic properties of these commutators. Theorem 3.40 Let Lß O Ÿ K. 1) ÒLß OÓ œ ÒOß LÓ 2) L and O commute elementwise if and only if ÒLß OÓ œ Ö"×, in particular, L Ÿ ^ÐKÑ

Í

ÒLß KÓ œ Ö"×

3) L üK

Í

ÒLß KÓ Ÿ L

4) If Lß O ü K, then ÒLß OÓ Ÿ L ∩ O

and

ÒLß OÓ ü K

Also, Lß O « K

Ê

ÒLß OÓ « K

5) L normalizes ÒLß OÓ and so ÒLß OÓ ü ØLß OÙ 6) LÒLß OÓ œ L O ü L O O œ ØLß OÙ In particular, ncÐLß KÑ œ LÒLß KÓ

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Fundamentals of Group Theory

Proof. We prove only part 5). Theorem 3.37 implies that if 2" − L , then Ò2ß 5Ó2" œ Ò2" 2ß 5ÓÒ2" ß 5Ó" − ÒLß OÓ Hence, L normalizes ÒLß OÓ. Similarly, O normalizes ÒLß OÓ and so ØLß OÙ also normalizes ÒLß OÓ.… Here are some additional properties of commutators of subgroups. Theorem 3.41 Let Eß Lß O Ÿ K. 1) ÒEß LOÓ œ ÒEß LÓÒEß OÓL where ÒEß OÓL œ ØÒ+ß 5Ó2 ± + − Eß 5 − Oß 2 − LÙ 2) If E ü K and if Y œ ÖL3 ± 3 − M× is a family of normal subgroups of K, then ÒEß 2L3 Ó œ 2ÒEß L3 Ó

This equation still holds even if one member of Y is not normal. In particular, if Eß L ü K and O Ÿ K, then ÒEß LOÓ œ ÒEß LÓÒEß OÓ 3) If R ü K, then ÒLR ß OR Ó Ÿ ÒLß OÓR and so ÒLR ß OR ÓR œ ÒLß OÓR 4) If R ü K, then ”

LR OR ÒLß OÓR ß •œ R R R

Proof. For part 1), if + − E, 2 − L and 5 − O , then Ò+ß 25Ó œ Ò+ß 2ÓÒ+ß 5Ó2 − ÒEß LÓÒEß OÓL and so ÒEß LOÓ Ÿ ÒEß LÓÒEß OÓL . For the reverse inclusion, we have ÒEß LÓ Ÿ ÒEß LOÓ and since Ò+ß 5Ó2 œ Ò+ß 2Ó" Ò+ß 25Ó − ÒEß LOÓ we also have ÒEß OÓL Ÿ ÒEß LOÓ.

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For part 2), note that part 1) and the fact that ÒEß LÓ ü K imply that ÒEß LOÓ Ÿ ÒEß LÓÒEß OÓ and the reverse inclusion is evident. For the general case, since ÒEß L5 Ó Ÿ ÒEß 1L3 Ó for all 5 , it follows that 2ÒEß L5 Ó Ÿ ÒEß 2L3 Ó

For the reverse inclusion, each generator of ÒEß 1L3 Ó belongs to a commutator subgroup of the form ÒEß L3" âL38 Ó, which is equal to 1ÒEß L34 Ó, which in turn is contained in 1ÒEß L3 Ó and so ÒEß 2L3 Ó Ÿ 2ÒEß L3 Ó

Part 3) follows from the fact that ÒLR ß OR Ó œ ÒLR ß OÓÒLR ß R ÓO œ ÒOß LÓÒOß R ÓL ÒLR ß R ÓO Ÿ ÒOß LÓR Part 4) follows from part 3).…

*Multivariable Commutators We can extend the definition of commutators as follows. If +ß ,ß - − K , then Ò+ß ,ß -Ó œ Ò+ß Ò,ß -ÓÓ and in general, if +" ß á ß +8 − K, then Ò+" ß á ß +8 Ó œ Ò+" ß Ò+# ß á ß +8 ÓÓ with Ò+Ó ³ +. Note that some authors define Ò+ß ,ß -Ó to be ÒÒ+ß ,Óß -Ó and, in general, these are not the same. Theorem 3.42 Let K be a group. The following are equivalent: 1) (Associativity) For all +ß ,ß - − K, ÒÒ+ß ,Óß -Ó œ Ò+ß Ò,ß -ÓÓ 2) (Distributivity) For all +ß ,ß - − K, Ò+ß ,-Ó œ Ò+ß ,ÓÒ+ß -Ó 3) (Commutator subgroup is central) Kw Ÿ ^ÐKÑ

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Proof. Since Ò+ß ,-Ó œ Ò+ß ,ÓÒ+ß -Ó, it follows that 2) holds if and only if Ò+ß -Ó, œ Ò+ß -Ó for all +ß ,ß - − K , which holds if and only if Ò+ß -Ó − ^ÐKÑ for all +ß - − K , that is, if and only if Kw Ÿ ^ÐKÑ. Hence, 2) and 3) are equivalent. It is clear that 3) implies 1). Conversely, 1) is Ò+ß ,Ó-Ò+ß ,Ó" - " œ +Ò,ß -Ó+" Ò,ß -Ó" which is equivalent to Ò+ß ,Ó-Ò,ß +Ó- " œ +Ò,ß -Ó+" Ò,ß -Ó" Now, as to the last factor Ò,ß -Ó" , note that 1) with , œ - implies that ÒÒ+ß ,Óß ,Ó œ " for all +ß , − K. Hence, Ò,ß -Ó" œ Ò-ß ,Ó œ ,Ò, " ß -Ó, " œ Ò, " ß -Ó and so 1) is equivalent to Ò+ß ,Ó-Ò,ß +Ó- " œ +Ò,ß -Ó+" Ò, " ß -Ó Expanding the commutators gives +,+" , " -,+, " +" - " œ +,-, " - " +" , " -,- " and cancelling gives +" , " -,+, " +" œ -, " - " +" , " -, Moving the first factor on the right side to the left side and the last factor on the left side to the right side gives - " +" , " -,+, " œ , " - " +" , " -,+ which is equivalent to Ò- " ß +" , " Ó, " œ , " Ò- " ß +" , " Ó which shows that Ò- " ß +" , " Ó commutes with , " , but - " ß +" , " and , " represent arbitrary elements of K and so ÒBß CÓ commutes with D for all Bß Cß D − K, that is, Kw Ÿ ^ÐKÑ.… Theorem 3.43 (Hall–Witt Identity) Let +ß ,ß - − K. Then Ò+ß , " ß -Ó, Ò,ß - " ß +Ó- Ò-ß +" ß ,Ó+ œ "

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Proof. For the first factor we have Ò+ß , " ß -Ó, œ ,Ò+ß Ò, " ß -ÓÓ, " œ ,+Ò, " ß -Ó+" Ò, " ß -Ó" , " œ ,+, " -,- " +" -, " - " ,, " œ ,+, " -,- " +" -, " - " œ +, , - +" Ð, " ÑBy cycling +, , and - , we get Ò,ß - " ß +Ó- œ , - - + , " Ð- " Ñ+ and Ò-ß +" ß ,Ó+ œ - + +, - " Ð+" Ñ, The product is easily seen to self-destruct.… If Lß O and P are subgroups of K, then we define ÒLß Oß PÓ œ ÒLß ÒOß PÓÓ and more generally, if L" ß á ß L8 Ÿ K, then ÒL" ß á ß L8 Ó œ ÒL" ß ÒL# ß á ß L8 ÓÓ with ÒLÓ ³ L . The subgroup ÒLß ÒOß PÓÓ is generated by elements of the form Ò2ß -" â-8 Ó, where 2 − L and -3 œ Ò53 ß j3 Ó for 53 − O and j3 − P. Hence, Theorem 3.37 implies that Ò2ß -" â-8 Ó œ $ Ò2ß -3 Óα3

where α3 − Ø2ß -" ß á ß -8 Ù Ÿ ØLß Oß PÙ. Corollary 3.44 (Three subgroups lemma) Let K be a group. Let L , O and P be subgroups of K. Then if any two of the commutators ÒLß Oß PÓ, ÒPß Lß OÓ or ÒOß Pß LÓ are contained in a normal subgroup R of K, then so is the third commutator. Proof. We assume that ÒLß Oß PÓ Ÿ R and ÒPß Lß OÓ Ÿ R and use the Hall– Witt Identity. Since Ò2ß 5 " ß jÓ − ÒLß Oß PÓ and Òjß 2 " ß 5Ó − ÒPß Lß OÓ for 2 − L , 5 − O and j − P, it follows from the Hall-Witt identity that Ò5ß j" ß 2Ój œ ÐÒ2ß 5 " ß jÓ5 Òjß 2 " ß 5Ó2 Ñ" − R and since R is normal in K, we deduce that Ò5ß j" ß 2Ó − R . Hence, ÒOß Pß LÓ Ÿ R .…

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Exercises 1.

Let K be a group and let L ü K. Prove that if L and KÎL are centerless, then K is centerless. 2. Let K œ W$ } G# . Show that K has two subgroups L and O of order ' that are centerless but that K œ LO is not centerless. Thus, the set product of centerless subgroups may be a group that is not centerless. What about the direct product of centerless groups? 3. If a group K has the property that all of its subgroups are normal, must K be abelian? 4. Let K be a finite group and let R" ß á ß R7 be normal subgroups of K . Show that kR" âR7 k divides kR" kâkR7 k. 5. A subgroup L of a group K is permutable if LO œ OL for all subgroups O of K. Prove that a maximal subgroup that is permutable is normal. 6. Show that if a normal subgroup L of a group K contains no nonidentity commutators, then L is central in K. 7. Use the Feit–Thompson Theorem to prove that a nontrivial group K of odd order is not perfect, that is, Kw  K. 8. Suppose that K is a finite group and that Kw  K. Prove that K has a normal subgroup O of prime index. 9. Show that the set product of subnormal subgroups need not be a subgroup. Hint: Check the dihedral group H) . 10. Let K be a finite group and let L be a subgroup for which ÐK À LÑ is prime. Show that if there is at least one left coset +L of L other than L itself that is also equal to some right coset L, , then L ü K . 11. a) Prove that the commutator subgroup of the quaternion group is Ö"ß "×. b) Prove that the commutator subgroup of the dihedral group H#8 is Hw#8 œ Ø3# Ù. 12. For which values of 8 is it true that the dihedral group H#8 has a pair of proper normal subgroups L and O for which H#8 œ LO

and L ∩ O œ Ö"×

13. Let K be a group and let L Ÿ K. Prove that GK ÐLÑ ü RK ÐLÑ. 14. Prove that W8w œ E8 for 8   % in the following manner. Use the fact that E8 is simple for all 8 Á %. Show that W8w ∩ E8 ü E8 . Deduce that W8w ∩ E8 œ Ö+× or W8w ∩ E8 œ E8 . Show that W8w ∩ E8 œ Ö+× is impossible. 15. Prove that if R ü K, then Œ 16. a)

K R

w

œ

Kw R R

Find an infinite group that is periodic. Hint: Look at the quotient groups of the additive group of rationals. b) Prove that if L ü K and L and KÎL are periodic, then so is K.

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17. Let R Ÿ ^ÐKÑ, the center of K. Show that R ü K and that if KÎR is cyclic, then K is abelian. 18. Prove that a maximal subgroup Q of a group K is normal if and only if Kw Ÿ Q . 19. Let L – K have prime index. Let B − L have the property that GL ÐBÑ  GK ÐBÑ. Prove that an element C − L is conjugate to B in K if and only if it is conjugate to B in L . 20. Let 1 œ Ö:" ß á ß :7 × be a nonempty set of primes. A 1-group is a group whose order 8 has the property that all primes dividing 8 lie in the set 1. For example, a group of order #$ † & † (# is a Ö#ß &ß (×-group. Let K be a finite group and let L" ß á ß L7 be normal subgroups such that KÎL3 is a 1-group for all 3. Prove that KÎ+L3 is also a 1-group. 21. A subgroup L of a group K is abnormal if + − ØLß L + Ù for all + − K. Prove that L is abnormal if and only if it satisfies the following conditions: a) If L Ÿ O Ÿ K, then RK ÐOÑ œ O . b) L is not contained in distinct conjugate subgroups, that is, if O Ÿ K and O Á O + , then L © y O ∩ O+. 22. Let Lß O Ÿ K. a) Prove that L O œ LÒLß OÓ ü ØLß OÙ œ L O O b) If K œ Ø\Ù for some subset \ of K, show that Kw œ ØÒBß CÓ ± Bß C − \Ùnor 23. Let W be a nonempty subset of a group K and let Lß O Ÿ K . Show that W LO œ ÐW O ÑL 24Þ Let Eß Lß O Ÿ K. Prove that ÒELß EOÓ œ ÒELß EÓÒELß OÓ 25. Let Lß Oß P Ÿ K . a) Show that if ÒLß Oß PÓ œ Ö"× and ÒPß Lß OÓ œ Ö"×, ÒOß Pß LÓ œ Ö"×. b) Show that if L , O and P are normal in K, then

then

ÒLß Oß PÓ Ÿ ÒPß Lß OÓÒOß Pß LÓ 26. Let K be a group and let R be a normal subgroup. Let Bß C − K and suppose that B8 C7 − R and CBC>+7 B"+8 − R . Prove that CBC> B" − R . 27. Let K be a group and let L Ÿ K. Prove that if L has finite index in K, then ÐK À LÑ œ ÐK À L B Ñ for any B − K.

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28. Let K be a nonabelian group. Show that ^ÐKÑ is a proper subgroup of any centralizer GK Ð1Ñ. 29. Let L Ÿ K. Prove that GK ÐLÑ ü RK ÐLÑÞ 30. Let K be a group and let L Ÿ K. a) Prove that R Ÿ K is normal in K if and only if all subsets of K normalize R . b) Prove that if W normalizes the subgroups L and O of K, then W normalizes their join L ” O , that is, RK ÐLÑ ∩ RK ÐOÑ Ÿ RK ÐL ” OÑ c)

Prove that if W and X normalize L , then the set product WX normalizes L. d) Prove that a subgroup L Ÿ K normalizes all supergroups of itself. 31. Prove that if L Ÿ K, then RK ÐL + Ñ œ RK ÐLÑ+ . In particular, GK Ð2Ñ+ œ GK Ð2+ Ñ. 32. Let K be a group and let L Ÿ O Ÿ K with L ü K. Prove that RK Œ

L O

œ

RK ÐLÑ O

33. a)

34.

35. 36. 37.

38.

Show that the subgroup L œ Ö+ß Ð" #ÑÐ$ %Ñß Ð" $ÑÐ# %Ñß Ð" %ÑÐ# $Ñ× of W% is normal. Hint: Show that the conjugate of any transposition is another transposition, in fact, Ð+ ,Ñ5 œ Ð5+ 5,Ñ. b) Show that the subgroup O œ Ö3ß Ð" #ÑÐ$ %Ñ× is normal in L but not normal in W% . c) Conclude that normality is not transitive. Let K be a group. a) Let Q be the intersection of all subgroups of K that have finite index in K. Show that Q is normal in K. b) Let L Ÿ K with finite index. Show that there is a normal subgroup R ü K for which R Ÿ L Ÿ K where R also has finite index in K. Let K be a group generated by two involutions B and C. Show that K has a normal subgroup of index #. Let K be a finite group of odd order. Show that the product of all of the elements of K, taken in any order, is in the commutator subgroup Kw . (P. Hall) Let Eß F Ÿ K. Show that if ÒEß Eß FÓ œ Ö"×, then ÒEß FÓ is an abelian group. Hint: One possible proof is as follows: Show that E commutes with ÒEß FÓ and with ÒFß EÓ and that F commutes with Ew . Then use a direct computation to show that Ò+ß ,ÓÒBß CÓ œ ÒCß B" ÓÒ+ß ,Ó where +ß B − E and ,ß C − F . Finally, use Theorem 3.37. Let : Á # be prime and consider the group K œ ™Ð:∞ Ñ { ™# where ™Ð:∞ Ñ is the :-quasicyclic group. Show that K has a unique

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maximal subgroup Q , but that Q is not maximum in the lattice of all proper subgroups of K. 39. A group K is said to be metabelian if its commutator subgroup Kw is abelian. (Some authors define a group K to be metabelian if Kw is central in K, which is stronger than our definition.) a) Prove that K is metabelian if and only if K has a normal abelian subgroup E for which KÎE is also abelian. b) Prove that the dihedral group H#8 is metabelian. c) Let J be a finite field, such as ™: where : is prime. Let +ß , − J where + Á !. The map 5+ß, À J Ä J defined by 5+ß, À B È +B  , is called an affine transformation of J . Show that the set affÐJ Ñ of all affine transformations of J is a subgroup of J J . Show that affÐJ Ñ is metabelian. d) Prove that if K œ EF where E and F are abelian, then K is metabelian. Hint: Show that Ò+ß ,Ó," +" œ Ò+ß ,Ó+" ," for +ß +" − E and ,ß ," − F . Use the fact that EF œ FE. 40. Let K be a group of order :8 , where : is prime. Suppose that for each + − K, the centralizer G+ œ GK Ð+Ñ has index " or :. a) Prove that G+ ü K. Hint: Assume ÐK À G+ Ñ œ :. Let M œ , G+1 Ÿ G+ 1−K

Show that M is normal in K. Then show that the elements of KÎM are actually permutations of KÎG+ , where 1MÐBG+ Ñ œ 1BG+ . Show that KÎM is a subgroup of WKÎL . b) Prove that Kw Ÿ ^ÐKÑ. 41. We have seen that the family subÐKÑ of all subgroups of K is a complete lattice. Let norÐKÑ be the family of normal subgroups of K. a) Show that the join of two normal subgroups E and F is the set product EF. b) Show that norÐKÑ is a complete sublattice of subÐKÑ. c) Prove that the distributive laws E ” ÐF ∩ GÑ œ ÐE ” FÑ ∩ ÐE ” GÑ E ∩ ÐF ” GÑ œ ÐE ∩ FÑ ” ÐE ∩ GÑ for Eß Fß G Ÿ K imply the modular law: For Eß Fß G Ÿ K with E Ÿ F, E ” ÐF ∩ GÑ œ F ∩ ÐE ” GÑ A lattice that satisfies the modular law is said to be a modular lattice. d) Prove that norÐKÑ is a modular lattice.

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Fundamentals of Group Theory e)

Is norÐKÑ necessarily distributive, that is, do the distributive laws necessarily hold? Hint: Consider the %-group Z œ Ö"ß +ß ,ß +,×. f) Prove that subÐKÑ need not be modular. Hint: Consider the alternating group E% of order "#. Let E œ ØÐ" #ÑÐ$ %ÑÙß F œ ØÐ" #ÑÐ$ %Ñß Ð" $ÑÐ# %ÑÙ and G œ ØÐ" # $ÑÙ. g) Prove the Dedekind law: For Eß Fß G Ÿ K with E Ÿ F , EÐF ∩ GÑ œ F ∩ ÐEGÑ Find an example to show that for arbitrary subgroups Eß Fß G of K, EÐF ∩ GÑ is not necessarily equal to EF ∩ EG and so the condition that E Ÿ F is necessary. h) Let Eß F and G be subgroups of K with E Ÿ F . Prove that if E∩G œF∩G

and

EG œ FG

then E œ F . 42. Let K be a group with subgroups R and L and suppose that R L Ÿ K. Prove that if ÐK À LÑ and kR k are finite and relatively prime, then R Ÿ L . 43. Let K be a group and L Ÿ K. a) Let L © \ © K. Show that the relation B ´ C if B" C − L is an equivalence relation on \ . What do the equivalence classes of this relation look like? Let Ð\ À LÑ be the cardinality of the set of equivalence classes. b) If L © \ © ] © K, where \L © \ . Show that if Ð\ À LÑ œ Ð] À LÑ  ∞, then \ œ ] . 44. Prove that R ü K and KÎR both have the ACC on subgroups if and only if K has the ACC on subgroups. 45. Sometimes it is useful to relate the commutator subgroup ÒLß OÓ of subgroups L and O to the commutator subgroup Ò\ß ] Ó of generating sets for L and O . Let \ and ] be nonempty subsets of a group K. a) Show that Ò\ß Ø] ÙÓ œ Ò\ß Ø] ÙÓØ] Ù œ Ò\ß ] ÓØ] Ù b) Show that ÒØ\Ùß Ø] ÙÓ œ Ò\ß ] ÓØ\ÙØ] Ù œ Ò\ß ] ÓØ] ÙØ\Ù

Complex Groups Let K be a group. Let Z be a nonempty family of subsets of K that forms a group under set product. Such a group has been called a complex group based on K (see Allen [1]). Let -Z be the union of the subsets in Z . We denote the identity of Z by I . Also, we denote the inverse of E − Z by E" . In the following exercises, let Z be an arbitrary complex group based on the group K.

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46. Let R ü K. Show that KÎR is a complex group in which the members form a partition of K . Thus, quotient groups are complex groups. 47. Show that if  is the multiplicative group of all positive rational numbers and if d œ ÖÐ<ß ∞Ñ ± < − ‘× then d is a complex group. Do the members of d form a partition of  ? What is the identity of d ? Does it contain the identity " −  ? Compare the sizes of  and d . Could d be a quotient group of  ? 48. Let ™ be the additive group of integers and let 5 be a positive integer. Show that the set m5 œ ÖÖ8× ∪ Ò8  5ß ∞Ñ ± 8 − ™× is a complex group. What is the identity of m5 ? What is the negative of the element E œ Ö"× ∪ Ò"  5ß ∞Ñ? Is this the set of negatives in ™ of the elements of E? 49. Show that all members of Z have the same cardinality. 50. Show that for Eß F − Z , E©F

Ê

F " © E"

51. Show that for Eß F − Z , I Ÿ Kß EF œ I

Ê

E∪F ©I

or ÐE ∪ FÑ ∩ I œ g

52. Show that for Eß F − Z , / − EF " ∩ FE"

Ê

EœF

and EF " ∪ FE" © I

Ê

EœF

53. Suppose that the members of Z form a partition of K. Prove that Z is a quotient group of K, that is, Z is the set of cosets of some normal subgroup of K. Hint: Show that " − I and that I ü K. 54. Prove that Z is a quotient group of K if and only if I Ÿ K . Hint: Prove that I Ÿ K if and only if the members of Z are pairwise disjoint and -Z Ÿ K. 55. Prove that EÁF−Z

Ê

E∩F œg

or E ∩ F is infinite

Chapter 4

Homomorphisms, Chain Conditions and Subnormality

Homomorphisms The structure-preserving functions between two groups are referred to as homomorphisms. Before giving a formal definition, let us make a few remarks about functions. We denote the action of a function 0 À W Ä X on = − W by either 0 = or 0 Ð=Ñ, depending on readability. If k denotes the power set, then the induced map 0 À kÐWÑ Ä kÐX Ñ is defined by 0 ÐY Ñ œ Ö0 Ð?Ñ ± ? − Y × and the induced inverse map 0 " À kÐX Ñ Ä kÐWÑ is defined by 0 " ÐZ Ñ œ Ö= − W ± 0 Ð=Ñ − Z × If 0 À W Ä W , then a subset E © W is invariant under 0 , or 0 -invariant, if 0 ÐEÑ © E. If Y is a family of functions from W to W , then a subset E © W is Y invariant if E is 0 -invariant for all 0 − Y . Now we can define homomorphisms. Definition Let K and L be groups. A function 5À K Ä L is called a group homomorphism (or just homomorphism) if 5Ð+,Ñ œ Ð5+ÑÐ5,Ñ The set of all homomorphisms from K to L is denoted by homÐKß LÑ. The following terminology is employed: 1) A surjective homomorphism is an epimorphism, which we denote by 5À K q» L .

S. Roman, Fundamentals of Group Theory: An Advanced Approach, DOI 10.1007/978-0-8176-8301-6_4, © Springer Science+Business Media, LLC 2012

105

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Fundamentals of Group Theory

2) An injective homomorphism is a monomorphism or embedding, which we denote by 5À K ä L . If there is an embedding from K to L , we say that K can be embedded in L and write K ä L . 3) A bijective homomorphism is an isomorphism, which we denote by 5À K ¸ L . If there is an isomorphism from K to L , we say that K and L are isomorphic and write K ¸ L . 4) A homomorphism of K into itself is an endomorphism. The set of all endomorphisms of K is denoted by EndÐKÑ. 5) An isomorphism of K onto itself is an automorphism. The set of all automorphisms of K is denoted by AutÐKÑ.… If 5À K Ä L is a homomorphism, then it is easy to see that 5" œ "

and 5Ð+" Ñ œ Ð5+Ñ"

for any + − K. Also, if 5À K ¸ L , then the inverse map 5" À L ¸ K is an isomorphism from L to K. The map that sends every element of K to the identity " − L is called the zero map. (We cannot call it the identity map!) In general, induced inverse maps are more well behaved than induced direct maps. Here is an example. Theorem 4.1 Let 5À K Ä L be a group homomorphism. 1) a) (Image preserves subgroups) WŸK

Ê

5W Ÿ L

b) (Surjective image preserves normality) If 5 is surjective, then W üK

Ê

5W ü L

2) (Inverse image preserves subgroups and normality) X ŸL

Ê

5" X Ÿ K

X üL

Ê

5 " X ü K

and …

While it is true that isomorphic groups have essentially the same group– theoretic structure, one must be careful in applying this notion to the subgroup structure of a group. For example, in the group ™ of integers, the subgroups #™ œ Ö#8 ± 8 − ™× and $™ œ Ö$8 ± 8 − ™× are isomorphic but the quotients ™Î#™ and ™Î$™ have different sizes. Thus, isomorphic subgroups need not have the same index. Example 4.2 Let H' œ Ø3ß 5Ù be the dihedral group, where 9Ð3Ñ œ $ and 9Ð5Ñ œ #. Let W$ be the symmetric group of order '. The map 0 À H' Ä W$ defined by

Homomorphisms, Chain Conditions and Subnormality

107

0 Ð53 35 Ñ œ Ð# $Ñ3 Ð" # $Ñ5 is an isomorphism and so H' ¸ W$ . This tells us that the group of symmetries of the triangle is the group of permutations of the vertices.… Definition Let K be a group and let 5À K Ä K be an endomorphism of K . 1) 5 is nilpotent if 58 œ ! for some 8  !, where ! is the zero map. 2) 5 is idempotent if 5# œ 5

…

We leave it as an exercise to show that the zero map is the only endomorphism that is both nilpotent and idempotent.

Sums of Homomorphisms If 5ß 7 À K Ä L are homomorphisms, then the map 5  7 À K Ä L defined by Ð5  7 ÑÐ+Ñ œ Ð5+ÑÐ7 +Ñ is a homomorphism if and only if the images imÐ5Ñ and imÐ7 Ñ commute elementwise. Definition Let K and L be groups. The sum of a family 5" ß á ß 58 À K Ä L of homomorphisms is the function defined by Ð5"  â  58 ÑÐ+Ñ œ Ð5" +ÑâÐ58 +Ñ for all + − K.… If the images imÐ53 Ñ commute elementwise, then the sum 5"  â  58 is a homomorphism and in this case, the sum itself is commutative, that is, 5"  â  58 œ 51"  â  518 for any 1 − W8 . Moreover, composition distributes over addition, 5Ð5"  â  58 Ñ œ 55"  â  558 and Ð5 "  â  5 8 Ñ 5 œ 5 " 5  â  5 8 5 Hence, if the images imÐ53 Ñ commute elementwise, then the binomial formula holds,

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Fundamentals of Group Theory

Ð5 "  â  5 8 Ñ7 œ 4" â48 œ7

Œ

7 54" â5848 4" ß á ß 48 "

for any 7  !. Theorem 4.3 Let 5" ß á ß 58 − EndÐKÑ have the property that the images imÐ53 Ñ commute elementwise. If each 53 is nilpotent, then so is the sum 7 œ 5 "  â  58 Proof. For any 7  !, 7 87 œ 4" â48 œ87

Œ

87 54" â5848 4" ß á ß 48 "

But if each 53 is nilpotent, then there is a positive integer 7 for which 537 œ ! for all 3 and so each term in the sum above is the zero map, since one of the exponents 43 is greater than or equal to 7.…

Kernels and the Natural Projection The kernel of a homomorphism 5À K Ä L is the subgroup kerÐ5Ñ œ Ö+ − K ± 5+ œ "× consisting of all elements of K that are sent to the identity of L . This is easily seen to be a normal subgroup of K. Conversely, any normal subgroup is a kernel. Theorem 4.4 Let R ü K. The map 1R À K Ä KÎR defined by 1R Ð+Ñ œ +R is an epimorphism with kernel R and is called the natural projection or canonical projection modulo R .… It is possible to tell whether a homomorphism is injective from its kernel. Theorem 4.5 A group homomorphism 5À K Ä L is injective if and only if kerÐ5Ñ œ Ö"×.…

Groups of Small Order One of the most important outstanding problems of group theory is the problem of classifying various types of groups up to isomorphism. This problem is called the classification problem and is, in general, very difficult. The classification problem for all groups is unsolved. The classification problem for finitelygenerated abelian groups is solved (see Theorem 13.4). The classification

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109

problem for finite simple groups seems to have been solved, and we discuss this in more detail in a later chapter. Groups of relatively small order have been fully classified up to isomorphism. For example, one can find such a classification of groups of order &! or less in Weinstein [36]. Of course, Lagrange's theorem gives a simple solution to the classification problem for groups of prime order, since all such groups are cyclic. A bit later in the book, we will solve the classification problem for groups of order "& or less. For now, we can solve the classification problem for groups of order ) or less, which are easily analyzed by looking at the possible orders of the elements. In fact, since groups of prime order are cyclic, we need only look at groups of order %, ' and ).

Groups of Order % Let K be a group of order %. If K has an element of order %, then K is cyclic and K ¸ G% . Otherwise, all nonidentity elements have order #, which implies that K is abelian. In this case, if +ß , − K are distinct nonidentity elements, then K œ Ö"ß +ß ,ß +,× is the Klein %-group. Thus, the groups of order % are (up to isomorphism): 1) G% , the cyclic group 2) Z ¸ G# } G# , the Klein 4-group.

Groups of Order ' If 9ÐKÑ œ ', then Cauchy's theorem implies that there exist +ß , − K with 9Ð+Ñ œ $ and 9Ð,Ñ œ #. Since Ø+Ù has index #, it is normal in K and since conjugation preserves order, we have +, œ +

or +, œ +#

If +, œ +, then K is abelian. In this case, since 9Ð+Ñ and 9Ð,Ñ are relatively prime, 9Ð+,Ñ œ 9Ð+Ñ9Ð,Ñ œ ' and so K is cyclic. If +, œ +# œ +" , then K œ Ø+ß ,Ùß 9Ð+Ñ œ $ß 9Ð,Ñ œ #ß ,+ œ +" , and so K is the dihedral group H' . Also, since H' ¸ W$ , the group K is also a symmetric group. Thus, the groups of order ' are (up to isomorphism): 1) G' , the cyclic group 2) H' ¸ W$ , the nonabelian dihedral (and symmetric) group.

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Fundamentals of Group Theory

Groups of Order ) Let 9ÐKÑ œ ). If K has an element of order ), it is cyclic and K ¸ G) . If every nonidentity element of K has order #, then K is abelian. In this case, let +ß , and - be distinct elements of K with - Á +, . It is easy to see that K œ Ö"ß +ß ,ß -ß +,ß +-ß ,-ß +,-× and that K ¸ G# } G# } G# . Now suppose that K has an element + of order % but no elements of order ). Then Ø+Ù ü K. For any ,  Ø+Ù, we have K œ Ø+ß ,Ù and since +, − Ø+Ù has order %, we must have ,+, " œ +

or ,+, " œ +$

If ,+, " œ +, then K is abelian. Moveover, K has an element - of order # that is not in Ø+Ù. To see this, note that if ,  Ø+Ù and 9Ð,Ñ œ %, then , # − Ø+Ù has order # and so , # œ +# . Hence, Ð+,Ñ# œ +# , # œ +% œ " and so - œ +, . Thus, K œ Ø+Ù ì Ø-Ù ¸ G% } G# On the other hand, suppose that ,+, " œ +$ œ +" . If there is a involution ,  Ø+Ù, then +, is also an involution and so Theorem 2.36 implies that Ø,ß +,Ù is dihedral of order #9Ð+Ñ œ ), that is, K œ Ø,ß +,Ù ¸ H) . Finally, if all elements ,  Ø+Ù have order %, then , # − Ø+Ù and so K œ Ø+ß ,Ùß + Á ,ß 9Ð+Ñ œ %ß , # œ +# ß ,+, " œ +$ which is the quaternion group. Thus, the groups of order ) are (up to isomorphism): 1) 2) 3) 4) 5)

G) , the cyclic group G% } G# , abelian but not cyclic G# } G# } G# , abelian but not cyclic H) , the (nonabelian) dihedral group U, the (nonabelian) quaternion group.

A Universal Property and the Isomorphism Theorems The following property is key to many other properties of group homomorphisms. Definition Let K be a group and let O ü K. Let Y ÐKà OÑ be the family of all pairs ÐLß 5À K Ä LÑ, where 5À K Ä L is a group homomorphism and O © kerÐ5Ñ.

Homomorphisms, Chain Conditions and Subnormality

u

G

111

S

∃!τ

σ H Figure 4.1

Referring to Figure 4.1, a pair ÐWß ?À K Ä WÑ − Y ÐKà OÑ is universal in Y ÐKà OÑ if for any pair ÐLß 5À K Ä LÑ − Y ÐKà OÑ, there is a unique group homomorphism 7 À W Ä L for which the diagram in Figure 4.1 commutes, that is, for which 7 ‰?œ5 The map 7 is called the mediating morphism for 5 and we say that 5 can be factored uniquely through ? or that 5 can be lifted uniquely to W .… Existence and uniqueness (up to isomorphism) of universal pairs is given by the following theorem. Theorem 4.6 (Universal pairs) Let K be a group and let O ü K. 1) (Existence) The pair ÐKÎOß 1O À K Ä KÎOÑ where 1O is the canonical projection modulo O is universal in Y ÐKà OÑ. The mediating morphism for 5À K Ä L is the map 7 À KÎO Ä L defined by 7 Ð1OÑ œ 51 Also, imÐ7 Ñ œ imÐ5Ñ

and kerÐ7 Ñ œ kerÐ5ÑÎO

2) (Uniqueness) Referring to Figure 4.2

πK

G

u

G/K

∃!σ S

Figure 4.2

∃!τ

112

Fundamentals of Group Theory

if ÐWß ?À K Ä WÑ is also universal in Y ÐKà OÑ, then the mediating morphisms 5 and 7 are inverse isomorphisms, whence W ¸ KÎO . Proof. For part 1), a mediating morphism for 5À K Ä L , if it exists, must satisfy 7 Ð1OÑ œ 51 for all 1 − K and so must be unique. But the condition O © kerÐ5Ñ implies that 7 is a well defined map and it is easy to see that it is a homomorphism. Finally, imÐ7 Ñ œ 7 ÐKÎOÑ œ 7 ‰ 1O ÐKÑ œ 5ÐKÑ œ imÐ5Ñ and 7 Ð+OÑ œ !

Í

7 ‰ 1O Ð+Ñ œ !

Í

5+ œ !

Í

+ − kerÐ5Ñ

and so kerÐ7 Ñ œ kerÐ5ÑÎO . For part 2), since ÐKÎOß 1O Ñ and ÐWß ?À K Ä WÑ are both universal, ? can be factored uniquely through 1O , that is, 7 ‰ 1O œ ? and 1O can be factored uniquely through ?, that is, 5 ‰ ? œ 1O Hence, Ð7 ‰ 5 Ñ ‰ ? œ ? But ? can be factored uniquely through itself and since + ‰ ? œ ?, it follows that 7 ‰ 5 œ +. Similarly, 5 ‰ 7 œ + and so 5 and 7 are inverse isomorphisms.… The following well-known results are direct consequences of the universal property of the pair ÐKÎLß 1L Ñ. Theorem 4.7 (The isomorphism theorems) Let K be a group. 1) (First isomorphism theorem) Every group homomorphism 5À K Ä L induces an embedding 5À KÎkerÐ5Ñ ä L of KÎkerÐ5Ñ defined by 5Ð1kerÐ5ÑÑ œ 51 and so K ¸ imÐ5Ñ kerÐ5Ñ

Homomorphisms, Chain Conditions and Subnormality

113

2) (Second isomorphism theorem) If Lß O Ÿ K with O ü K , then L ∩ O ü L and LO L ¸ O L ∩O 3) (Third isomorphism theorem) If L Ÿ O Ÿ K with Lß O ü K, then OÎL ü KÎL and K O K „ ¸ L L O Hence ÐK À OÑ œ ÐKÎL À OÎLÑ Proof. For part 1), since ÐKÎOß 1O Ñ is universal, there is a unique homomorphism 7 À KÎO Ä imÐ5 Ñ for which 7 ‰ 1O œ 5. The rest follows from the fact that imÐ7 Ñ œ imÐ5Ñ and kerÐ7 Ñ œ kerÐ5ÑÎO œ Ö"×. For part 2), the map 5À L Ä LOÎO defined by 5Ð2Ñ œ 2O is clearly an epimorphism with kernel L ∩ O and the first isomorphism theorem completes the proof. Proof of part 3) is left to the reader.… Theorem 4.8 Let K" and K# be groups and let L3 ü K3 for 3 œ "ß #. Then K" } K# K" K# ¸ } L" } L# L" L# Proof. Let 7 À K" } K# Ä ÐK" ÎL" Ñ } ÐK# ÎL# Ñ be defined by 7 Ð+" ß +# Ñ œ Ð+" L" ß +# L# Ñ We leave it to the reader to show that 7 is an epimorphism with kernel L" } L# . The first isomorphism theorem then completes the proof.…

The Correspondence Theorem The following correspondence theorem has a great many uses. We use the notation subÐR à KÑ to denote the lattice of all subgroups of K that contain R . Theorem 4.9 (Correspondence theorem) Let K be a group, let R ü K and let 1À K Ä KÎR be the natural projection. Referring to Figure 4.3,

114

Fundamentals of Group Theory

G

π G/N

K

π K/N

H

π H/N

N

π {N} Figure 4.3

let 1À subÐR à KÑ Ä subÐKÎR Ñ be the map defined by 1ÐLÑ œ LÎR 1) 1 is an order isomorphism, that is, 1 is a bijection for which L ŸO

Í

LÎR Ÿ OÎR

for all Lß O − subÐR à KÑ. In particular, every subgroup of KÎR has the form LÎR for a unique L − subÐR à KÑ. 2) Normality is preserved in both directions, that is, L üO

Í

LÎR ü OÎR

for all Lß O − subÐR à KÑ. Moreover, the corresponding factor groups are isomorphic, that is, O O L ¸ „ L R R and so 1 also preserves index: ÐO À LÑ œ ÐOÎR À LÎR Ñ It follows that subnormality is also preserved in both directions: L üü O

Í

LÎR ü ü OÎR

3) If R « K, then the property of being characteristic is preserved in only one direction, specifically, L K « R R

Ê

L «K

for any L − subÐR à KÑ, but the converse fails. Proof. For the surjectivity of 1, any subgroup of KÎR has the form W œ ÖBR ± B − \×

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for some index set \ . The set L œ -ÐBR Ñ is a subgroup of K since Bß C − L implies that BR ß CR − W and so B" R and BCR are in W , whence B" ß BC − L . Also, 1L œ LÎR œ ÖBR ± B − L× œ W and so 1 is surjective. For part 3), if 5 − AutÐKÑ, then 1R ‰ 5À K Ä KÎR is an epimorphism with kernel 5" R œ R . Hence, the map 7 À KÎR Ä KÎR defined by 7 Ð+R Ñ œ Ð5+ÑR is an automorphism of KÎR and so 7 ÐLÎR Ñ Ÿ LÎR . Thus, for 2 − L , Ð52ÑR œ 7 Ð2R Ñ − LÎR and so 52 − L , which shows that L « K. As to the converse, let K œ O œ H) , L œ Ø3Ù and R œ ^ÐKÑ œ Ø3# Ù. Then L and R are characteristic in H) but LÎ^ÐKÑ ¸ G# is not characteristic in H) Î^ÐKÑ ¸ G# } G# . The rest of the proof is left to the reader.…

Group Extensions It will be convenient to make the following definition. Definition Let K be a group. We refer to subgroups L and O of K for which L Ÿ O as an extension. We also refer to L ü K as a normal extension. The index of an extension L Ÿ O is ÐO À LÑ.… This use of the term extension is consistent with its use in field theory, where if J © O are fields, then O is referred to as an extension of J . The term extension has another meaning in group theory, which we will encounter later in the book: An extension of a pair ÐR ß UÑ of groups is a group K that has a normal subgroup R w isomorphic to R and for which KÎR w ¸ U. However, since L Ÿ K is an extension whereas ÐR ß UÑ has an extension, there should be no ambiguity in adopting the current definition. Various operations can be performed on a group extension to yield another extension. Definition Let K be a group. Let E Ÿ F Ÿ K and G Ÿ K. 1) The intersection of E Ÿ F with G is E∩G ŸF∩G 2) If G ü K, the normal lifting of E Ÿ F by G is EG Ÿ FG

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3) If R Ÿ E Ÿ F with R ü F , then the quotient of E Ÿ F by R is E F Ÿ R R and (for want of a better term) the unquotient of EÎR Ÿ FÎR is E Ÿ F.…

Inheritance of Group Properties Let c be a property of groups, such as being cyclic, being finite or being abelian. (Technically, c can be thought of as a subclass of the class of all groups.) We write K − c to denote the fact that the group K has property c . A group property c is isomorphism invariant if K − cß

L ¸K

Ê

L−c

We will say that a normal extension E ü F has property c if the quotient FÎE has property c . For example, to say that E ü F is cyclic is to say that FÎE is cyclic. A property c of groups is inherited by subgroups if K − cß

L ŸK

Ê

L−c

and c is inherited by quotients if K − cß

L üK

Ê

KÎL − c

Preservation of Group Properties The second isomorphism theorem implies that intersection preserves normality, that is, E ü Fß

G ŸK

Ê

E∩G üF∩G

and that the quotient satisfies F∩G F∩G EÐF ∩ GÑ F œ ¸ Ÿ E∩G E ∩ ÐF ∩ GÑ E E Hence, any isomorphism-invariant property of E ü F that is inherited by subgroups, such as being finite, abelian or cyclic, is inherited by intersections. For example, if E ü F is cyclic, then so is E ∩ G ü F ∩ G . Similar statements can be made about normal liftings, quotients and unquotients, as follows. Theorem 4.10 Let K be a group with E Ÿ F Ÿ K. Let c be an isomorphisminvariant property of groups. 1) (Intersection) Normality is preserved by intersection, that is, for G Ÿ K , EüF

Ê

E∩G üF∩G

Also, if c is inherited by subgroups, then c is preserved by intersection.

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2) (Normal lifting) Normality is preserved by normal lifting, that is, for R ü K, EüF

Ê

ER ü FR

Also, if c is inherited by quotients, then c is preserved by normal lifting. 3) (Quotient and unquotient) Normality is preserved by quotient and unquotient, that is, for R ü F and R Ÿ E Ÿ F , EüF

Í

EÎR ü FÎR

Also, c is preserved by quotient and unquotient. Proof. For part 2), to see that ER ü FR , note that F normalizes both E and R and so F normalizes ER . Also, R Ÿ ER implies that R normalizes ER and so FR normalizes ER , that is, ER ü FR . Since FR œ EFR œ ER F , it follows that FR ER F F F F EÐF ∩ R Ñ œ ¸ œ ¸ ‚ ER ER F ∩ ER EÐF ∩ R Ñ E E

…

Centrality Another very important property of extensions E Ÿ F in a group K that involves more than just the quotient FÎE alone is centrality. Definition Let E ü K. The extension E ü F in K is central in K if F K Ÿ ^Œ E E

…

Note that centrality is not a property of FÎE alone, since it depends on KÎE as well, which is why we use the phrase central in K. Theorem 3.40 implies that an extension E ü F is central in K if and only if ÒFß KÓ Ÿ E Thus, for R ü K and R Ÿ E, E ü F central in K

Í

E F K ü central in R R R

and so centrality is preserved by quotient and unquotient. Also, for any L Ÿ K, ÒF ∩ Lß LÓ Ÿ E ∩ L and so E ü F central in K Finally, for any R ü K,

Ê

E ∩ L Ÿ F ∩ L central in L

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ÒFR ß KÓ œ ÒFß KÓÒR ß KÓ Ÿ ER and so E ü F central in K

Ê

ER Ÿ FR central in K

Thus, centrality is preserved by intersection and normal lifting as well. Theorem 4.11 Let K be a group. The property of an extension being central in K is preserved by intersection, normal lifting, quotient and unquotient.…

Projections and the Zassenhaus Lemma Combining intersection with lifting allows us to project one normal extension in K into another. Specifically to project E ü F into L ü O , we first intersect E ü F with O to get ÐE ∩ OÑ ü ÐF ∩ OÑ and then lift by L to get LÐE ∩ OÑ ü LÐF ∩ OÑ This extension is the projection of E ü F into L ü O and is denoted by ÐE ü FÑ Ä ÐL ü OÑ We leave it to the reader to show that the same extension is obtained by first lifting E ü F by L and then intersecting with O . As to the factor group of the projection, the isomorphism theorems give LÐF ∩ OÑ LÐE ∩ OÑÐF ∩ OÑ œ LÐE ∩ OÑ LÐE ∩ OÑ F∩O ¸ ÐF ∩ OÑ ∩ LÐE ∩ OÑ F∩O œ ÐE ∩ OÑÒÐF ∩ OÑ ∩ LÓ F∩O œ ÐE ∩ OÑÐF ∩ LÑ But the last quotient remains unchanged if we reverse the roles of the two extensions E Ÿ F and L Ÿ O . Hence, the reverse projection ÐL ü OÑ Ä ÐE ü FÑ has an isomorphic quotient, that is, LÐF ∩ OÑ EÐO ∩ FÑ ¸ LÐE ∩ OÑ EÐL ∩ FÑ

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This result was proved by Zassenhaus in 1934 and was given the name butterfly lemma by Serge Lang because of the shape of a certain figure associated with an alternate proof. Theorem 4.12 (Zassenhaus lemma [37], 1934) Let K be a group and let EüF

and

LüO

be normal extensions in K. Then the reverse projections ÐE ü FÑ Ä ÐL ü OÑ and

ÐL ü OÑ Ä ÐE ü FÑ

have isomorphic factor groups, that is, LÐF ∩ OÑ EÐO ∩ FÑ ¸ LÐE ∩ OÑ EÐL ∩ FÑ

…

Let us make a few very simple observations about projections: 1) If E ü F ü G , then the projections of the contiguous extensions E ü F and F ü G into L ü O are also contiguous, that is, LÐE ∩ OÑ ü LÐF ∩ OÑ ü LÐG ∩ OÑ 2) If E ü F is projected into L ü O where F   O , then the projection has top group subgroup O . 3) If E ü F is projected into L ü O where E Ÿ L , then the projection has bottom subgroup L . Thus, we can project a series E" ü E# ü â ü E8 in K into an extension L ü O to get a new series LÐE" ∩ OÑ ü LÐE# ∩ OÑ ü â ü LÐE8 ∩ OÑ In particular, if E" Ÿ L and O Ÿ E8 , then the series runs from L to O .

Inner Automorphisms If K is a group, then the map # À K Ä AutÐKÑ defined by # + œ #+ is a group homomorphism, since #+ #, œ #+, . The kernel of # is ^ÐKÑ and the image of # is InnÐKÑ, which is normal in AutÐKÑ since for any 5 − AutÐKÑ and 1 − K, Ð5#+ 5 " Ñ1 œ 5Ò+Ð5 " 1Ñ+" Ó œ Ð5+Ñ1Ð5+Ñ" œ #5+1 and so

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Fundamentals of Group Theory

#+5 œ #5+ An automorphism of K that is not an inner automorphism is called an outer automorphism of K and the factor group AutÐKÑÎInnÐKÑ is called the outer automorphism group of K, even though its elements are not automorphisms. Theorem 4.13 Let K be a group. 1) The map #À K Ä AutÐKÑ defined by # + œ #+ is a group homomorphism with image InnÐKÑ ü AutÐKÑ and kernel ^ÐKÑ. Hence, InnÐKÑ ¸

K ^ÐKÑ

In particular, if K is centerless, then K ¸ InnÐKÑ. 2) If L Ÿ K, then + − RK ÐLÑ

Í

#+ − AutÐLÑ

Moreover, the restricted map #À RK ÐLÑ Ä AutÐLÑ has kernel GK ÐLÑ and so GK ÐLÑ ü RK ÐLÑ and RK ÐLÑ ä AutÐLÑ GK ÐLÑ In particular, if L ü K, then GK ÐLÑ ü K and K ä AutÐLÑ GK ÐLÑ

…

Characteristic Subgroups Normality is not transitive: If R ü K, then the normal subgroups of R are not necessarily normal in K . Examples can be found in the symmetric group W% . However, the property of being characteristic is transitive. Here are some of the basic properties associated with characteristic subgroups. Theorem 4.14 Let K be a group. 1) L « K if and only if 5L œ L for all 5 − AutÐKÑ. 2) (Transitivity) E«F«G

Ê

E«G

E«FüG

Ê

EüG

and

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3) Any subgroup of a cyclic group is characteristic and so L Ÿ Ø+Ù ü K

Ê

L üK

4) (Extension property) For any R Ÿ L Ÿ K, R « Kß

L K « R R

Ê

L «K

Proof. For part 3), a finite cyclic group K has only one subgroup of each size and so it must be invariant under any automorphism of K. On the other hand, if K œ Ø+Ù is infinite cyclic and 5 − AutÐKÑ, then 5+ must be a generator of K and so 5+ œ + or 5+ œ +" . Hence, 5 is either the identity map or the map that sends any element to its inverse. In either case, any subgroup of K is 5 invariant.… Definition A nontrivial group K is characteristically simple if it has no nontrivial proper characteristic subgroups.… As an example, the %-group Z œ Ö"ß +ß ,ß +,× is characteristically simple but not simple.

Elementary Abelian Groups The simplest type of abelian group is a cyclic group of prime order. Perhaps the next simplest type of abelian group is an external direct product of cyclic groups of the same prime order. Definition An elementary abelian group K is an abelian group in which every nonidentity element has the same finite order.… Theorem 4.15 Let K be an elementary abelian group. 1) Every nonidentity element of K has prime order :. 2) Writing the group product additively, K is a vector space over ™: ,where if α − ™: and + − K, then α+ is the sum of α copies of +. The subgroups of K are the same as the subspaces of K and the group endomorphisms of K are the same as linear operators on K. 3) If K is finite, then K ¸ ™: } â } ™ : (This also holds when K is infinite, but we have not yet defined infinite direct products.) Proof. We use additive notation for K . For part 1), if 9Ð+Ñ œ 78 where 7  ", then 9Ð8+Ñ œ 7 and so 78 œ 7, that is, 8 œ ". Thus, 9Ð+Ñ œ : is prime for all nonidentity + − K . For part 2), let Š and  denote addition and multiplication in ™: , respectively. If 8 − ™ is nonnegative, let 8 ‡ + be the sum of 8 copies of +. Then for αß " − ™: , there exists an integer 8 for which

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Fundamentals of Group Theory

α Š " œ Ðα  " Ñ  8: and so Ðα Š " Ñ+ œ ÒÐα  " Ñ  8:Ó ‡ + œ Ðα  " Ñ ‡ + œ α+  ", Similarily, Ðα  " Ñ+ œ Òα"  8:Ó ‡ + œ Ðα" Ñ ‡ + œ αÐ" +Ñ The other requirements of scalar multiplication are also met, namely, "+ œ + and αÐ+  ,Ñ œ α+  α, We leave proof of the remaining part of part 2) to the reader. For part 3), K is vector-space isomorphic to a direct sum of a certain number of one-dimensional subspaces of K, that is, subspaces of the form ™: + œ Ø+Ù for + − K . These subspaces are cyclic subgroups of K and the vector space isomorphism is also a group isomorphism.… Theorem 4.16 The following are equivalent for an abelian group K: 1) K is elementary abelian 2) K is characteristically simple and has at least one nonidentity torsion element. Proof. If K is an elementary abelian group, then the automorphisms of K are the linear automorphisms of K . It follows that if W Ÿ K is nontrivial and proper, then for any nonzero + − W and nonzero B − K Ï W , there is an automorphism sending + to B and so W is not characteristic in K. Hence, K is characteristically simple. Of course, K has a nonidentity torsion element. For the converse, assume that K is characteristically simple and let : be a prime dividing the order of some element + − E. Then (using additive notation) E: œ Ö+ − E ± :+ œ !× is nontrivial and characteristic in E, whence E œ E: is an elementary abelian group.… Note that the :-quasicyclic group ™Ð:∞ Ñ, which has proper subgroup lattice Ö!×  Ø"Î:Ù  Ø"Î:# Ù  â is characteristically simple, since there is at most one subgroup of any given size. However, it is not an elementary abelian group since it has no nonzero torsion elements.

Homomorphisms, Chain Conditions and Subnormality

123

Multiplication as a Permutation If K is a group, then multiplication by + − K is a permutation of K; specifically, the multiplication map .+ À K Ä K defined by .+ B œ +B is bijective. In this context, multiplication is also referred to as (left) translation. The map .À K Ä WK that sends + to .+ provides a representation of the elements of the group K as permutations of the set K. Moreover, the representation . is a group homomorphism, since .+, œ .+ ., This is not the only way to represent the elements of a group as permutations of some set. For example, if L Ÿ K, then the elements of K can also be represented as permutations of a quotient set KÎL via the multiplication map 5+ À K Ä KÎL defined by 5+ Ð1LÑ œ +1L It is clear that 5+ is a permutation of KÎL and that the map 5À K Ä WKÎL sending + to 5+ is a group homomorphism. The representation map .À K Ä WK is described by saying that K acts on itself by (left) translation and the representation map 5À K Ä KÎL is described by saying that K acts on KÎL by (left) translation. Both of these representations fit the pattern of the following definition. Definition An action of a group K on a nonempty set \ is a group homomorphism -À K Ä W\ , called the representation map for the action. Thus, -Ð"Ñ œ +ß

-Ð+" Ñ œ Ð-+Ñ"

and

-Ð+,Ñ œ -Ð+Ñ-Ð,Ñ

for all +ß , − K. When - is a group action, we say that K acts on \ by -. The permutation -+ is usually denoted by -+ , or simply by + itself.… We should mention that translation is not the only important action of a group on a set. We will see in a later chapter that conjugation is also an important group action. Since a representation map is assumed only to be a homomorphism, it is possible for two distinct elements of K to have the same representation in W\ . Although it may seem at first that this is not a particularly desirable quality, we will see that such representations can yield important results. An injective representation, that is, an embedding -À K ä W\ is said to be faithful. In this

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Fundamentals of Group Theory

case, K is isomorphic to a subgroup of W\ . The action of K on itself by translation is faithful but the action of K on KÎL by translation need not be faithful. Here is one interesting application of left translation. Theorem 4.17 Let L be a proper subgroup of a group K. Then there are distinct group homomorphisms 5ß 7 À K Ä O into some group O that agree on L. Proof. For B  KÎL , we construct two distinct group actions of K on the set \ œ KÎL ∪ ÖB× that agree on L . Let 5À K Ä W\ be left translation on KÎL that also leaves B fixed, that is, for any + − K, 5+ Ð,LÑ œ +,L

and 5+ B œ B

and let 7 À K Ä W\ be defined by 7+ œ ÐL BÑ5+ ÐL BÑ If 2 − L , then 52 fixes both L and B and so 52 and ÐL BÑ are disjoint permutations, whence 52 œ 72 for all 2 − L . On the other hand, 7 and 5 are distinct since if +  L , then 7+ L œ L

but 5+ L Á L

…

The Left Regular Representation: Cayley's Theorem The action .À K Ä WK of K on itself by left translation is called the left regular representation of K. Since the left regular representation is faithful, it follows that every group is isomorphic to a subgroup of some symmetric group! Theorem 4.18 (Cayley's theorem [7], 1854) Every group K is isomorphic to a subgroup of the symmetric group WK , via the action of K on itself by left translation.…

Multiplication by K on KÎL ; the Normal Interior If L  K has finite index, the action 5À K Ä WKÎL of K on KÎL by left translation yields a variety of remarkable consequences, mainly due to the nature of the kernel of this action, which is the intersection of all conjugates of L:

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125

kerÐ5Ñ œ ÖB − K ± 5B œ +× œ ÖB − K ± B+L œ +L for all + − K× œ ÖB − K ± B+ − +L for all + − K× œ ÖB − K ± B − +" L+ for all + − K× œ , L+ +−K

Thus,

O ³ ,+−K L +

is both normal in K and contained in L . Moreover, if R is any normal subgroup of K that is contained in L , then R œ R + Ÿ L+ for all + − K and so R Ÿ O . In other words, O is the largest subgroup of L that is normal in K. Theorem 4.19 Let L Ÿ K. The largest subgroup of L that is normal in K is called the normal interior or core of L , which we denote by L ‰ . The core of L is the kernel L‰ œ , L+ +−K

of the action of K on KÎL by left translation.… Thus, left translation 5À K Ä WKÎL induces an embedding K ä WKÎL L‰ of KÎL ‰ into WKÎL and so ÐK À L ‰ Ñ ± ÐK À LÑx In particular, if K is simple, then L ‰ is trivial and so 9ÐKÑ ± ÐK À LÑx These facts have some rather interesting consequences relating to the existence of normal subgroups and subgroups of small index of a group. Theorem 4.20 Let K be a group and let L Ÿ K have finite index. Then KÎL ‰ ä WKÎL and so

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ÐK À L ‰ Ñ ± ÐK À LÑx In particular, ÐK À L ‰ Ñ is also finite and ÐL À L ‰ Ñ ± ÐÐK À LÑ  "Ñx 1) Any of the following imply that L ü K: a) L is periodic and ÐK À LÑ œ : is equal to the smallest order among the nonidentity elements of L . b) K is finite and 9ÐLÑ and ÐÐK À LÑ  "Ñx are relatively prime, that is, for all primes :, : ± 9ÐLÑ

Ê

:   ÐK À LÑ

which happens, in particular, if ÐK À LÑ is equal to the smallest prime dividing 9ÐKÑ. 2) If K is finitely generated, then K has at most a finite number of subgroups of any finite index 7. 3) If K is simple, then 9ÐKÑ ± ÐK À LÑx a) If K is infinite, then K has no proper subgroups of finite index. b) If K is finite and 9ÐKÑ y± 7x for some integer 7, then K has no subgroups of index 7 or less. Proof. For part 1), first note that : is a prime. If ; is a prime dividing ÐL À L ‰ Ñ, then Cauchy's theorem implies that there is an 2 − L for which ; œ 9Ð2L ‰ Ñ ± 9Ð2Ñ, whence ;   :, a contradiction to the fact that ; ± ÐL À L ‰ Ñ ± Ð:  "Ñx Hence, ÐL À L ‰ Ñ œ " and L œ L ‰ ü K. For part 2), ÐL À L ‰ Ñ divides both ÐÐK À LÑ  "Ñx and 9ÐLÑ and so must equal " ÐL À L ‰ Ñ ± Ð7  "Ñx But ÐL À L ‰ Ñ also divides 9ÐLÑ, which is relatively prime to Ð:  "Ñx and so ÐL À L ‰ Ñ œ ". For part 3), suppose that the result is true for all normal subgroups. Then there are only finitely many possibilities for normal interiors of subgroups of index 7, since such a normal interior must have index dividing 7x. But each normal subgroup R of finite index can be the normal interior for only finitely many subgroups, since there are only finitely many subgroups containing R . Now, normal subgroups correspond to kernels of homomorphisms. In particular, if ÐK À R Ñ œ 5 , then the homomorphism

Homomorphisms, Chain Conditions and Subnormality

127

5 ‰ 1R À K Ä KÎR ä W5 where 5 is the left regular representation of KÎR in W5 , has kernel R . Thus, distinct normal subgroups R of index 5 yield distinct homomorphisms from K into W5 . However, since K is finitely generated, there are only a finite number of such homomorphisms and so there are only a finite number of normal subgroups of K of index 5 .… Example 4.21 If 9ÐKÑ œ :8 ? with : prime and :  ?, then we will prove in a later chapter that K has a subgroup L of order :8 (a Sylow :-subgroup of K). Since :8 and Ð?  "Ñx are relatively prime, it follows that L ü K .…

The Frattini Subgroup of a Group The following subgroup of a group is most interesting. Definition The Frattini subgroup FÐKÑ of a finite group K is the intersection of the maximal subgroups of K.… Definition Let K be a group. An element + − K is called a nongenerator of K if whenever the subset W © K generates K, then so does the set W Ï Ö+×. Thus, a nongenerator is an element that is not needed in any generating set.… Here are the basic properties of the Frattini subgroup of a finite group. But first a definition. Definition Let K be a group and let O Ÿ K. A subgroup L of K is called a supplement of O if K œ LO .… Theorem 4.22 Let K be a finite group. 1) FÐKÑ is the set of all nongenerators of K. 2) FÐKÑ « K. 3) The following are equivalent: a) Every maximal subgroup of K is normal. b) KÎFÐKÑ is abelian. 4) If O – K, then O Ÿ FÐKÑ if and only if O has no proper supplements. Proof. For part 1), if W © K does not generate K, then W is contained in a maximal subgroup Q , which also contains FÐKÑ and so W ∪ FÐKÑ © Q does not generate K. Hence, the elements of FÐKÑ are nongenerators. Conversely, if B is a nongenerator of K and B  Q for some maximal subgroup Q , then Q ∪ ÖB× generates K and so Q generates K, which is false. Thus, all nongenerators of K are in FÐKÑ. For part 2), if 5 − AutÐKÑ, then the induced map is a bijection on the family ` of maximal subgroups of K and so

128

Fundamentals of Group Theory 5ÐFÐKÑÑ œ 5Š,`‹ œ ,5` œ ,` œ FÐKÑ

For part 3), assume that every maximal subgroup Q of K is normal. Then KÎQ has no proper nontrivial subgroups and so is abelian. Hence, Kw Ÿ Q and so Kw Ÿ FÐKÑ. Conversely, if Kw Ÿ FÐKÑ and Q is a maximal subgroup of K, then Kw Ÿ Q and so Q ü K. For part 4), assume first that O Ÿ FÐKÑ. If L  K , then L Ÿ Q for some maximal subgroup Q of K and so LO Ÿ Q  K. Conversely, if O has no proper supplements, then every maximal subgroup Q satisfies Q Ÿ Q O  K and so Q œ Q O , whence O Ÿ Q and so O Ÿ FÐKÑ.…

Subnormal Subgroups We wish now to take a closer look at the concept of subnormality. As to the existence of subnormal subgroups, we have the following examples to show that all possibilities may occur, and that a subnormal subgroup need not be normal. Example 4.23 1) A simple group has no nontrivial proper subnormal subgroups. 2) In the dihedral group H) œ Ø5ß 3Ù, we have Ø5 Ù – Ø 5 ß 3 # Ù – H ) and so Ø5Ù is subnormal in H) but not normal in H) . In fact, all subgroups of H) are subnormal. 3) In the symmetric group W$ , the subgroup L œ ØÐ" #ÑÙ is maximal and so the only sequence of subgroups from L to W$ is L Ÿ W$ . Since L is not normal in W$ , it follows that L is not subnormal in W$ . Of course, ØÐ" # $ÑÙ is subnormal in W$ , being normal in W$ .… The following theorem outlines the simplest properties of subnormality and is a direct consequence of Theorem 4.10. Theorem 4.24 Let K be a group. Let Lß O Ÿ K. 1) (Transitivity) L üü O

and

O üü K

Ê

L üü K

2) (Intersection) If P Ÿ K, then L üüO

Ê

L ∩PüüO ∩P

In particular, L Ÿ O Ÿ Kß

L üüK

Ê

L üüO

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129

and L ü ü Kß

O üüK

Ê

L ∩O üüK

3) (Normal lifting) If R ü K, then L üü O

Ê

LR ü ü OR

4) (Quotient/unquotient) If R Ÿ L and R ü O , then L üü O

Í

LÎR ü ü OÎR

…

Intersections and Subnormality While the intersection of two, and hence any finite number, of subnormal subgroups is subnormal, this is not the case for an arbitrary family of subnormal subgroups. Example 4.25 Let K be the infinite dihedral group, K œ Ö33 ß 533 ± 3 − ™× where 9Ð5Ñ œ #, 9Ð3Ñ œ ∞ and 35 œ 53" . If 3

L3 œ Ø5ß 3# Ù

then L3" ü L3 and so L3 ü ü K. But, +L3 œ Ø5Ù is self-normalizing (equal to its own normalizer) and so is not subnormal in K.… However, since the family subnÐKÑ is closed under finite intersection, Theorem 1.6 does imply the following. Theorem 4.26 If a group K has the DCC on subnÐKÑ, then the intersection of any family of subnormal subgroups is subnormal.…

The Sequence of Normal Closures of a Subgroup If L ü ü K, then the first step down in a series for L is an extension L" ü K. Moreover, since ncÐLß KÑ Ÿ L" , the largest possible first step down in a series from L to K is ncÐLß KÑ ü K By repeatedly taking normal closures of L , we get a series from L to K that descends more rapidly than any other series from L to K . Definition Let K be a group and let L Ÿ K. The sequence of normal closures of L in K is defined by

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Fundamentals of Group Theory

LÐ!Ñ œ Kß LÐ"Ñ œ ncÐLß LÐ!Ñ Ñ

and LÐ3"Ñ œ ncÐLß LÐ3Ñ Ñ

for 3   !.… To see that the sequence of normal closures descends more rapidly than any other proper series L œ L< – L<" – â – L! œ K it is clear that LÐ!Ñ Ÿ L! and if LÐ3Ñ Ÿ L3 , then LÐ3"Ñ œ ncÐLß LÐ3Ñ Ñ Ÿ ncÐLß L3 Ñ Ÿ L3" Hence, LÐ3Ñ Ÿ L3 for all 3. Theorem 4.27 A subgroup L of K is subnormal in K if and only if the sequence of normal closures of L in K reaches L , in which case this sequence is a series of shortest length from L to K . If L ü ü K , then the length of the sequence of normal closures is called the subnormal index of L in K, which we denote by =ÐLß KÑ.… We gather a few facts about subnormal indices. Theorem 4.28 Let K be a group and let L ü ü K have sequence of normal closures L œ LÐ=Ñ – LÐ="Ñ – â – LÐ!Ñ œ K 1) (Triangle inequality) L üü O üü K

Ê

=ÐLß KÑ Ÿ =ÐLß OÑ  =ÐOß KÑ

2) If 5 − AutÐKÑ, then 5ÐLÐ3Ñ Ñ œ Ð5LÑÐ3Ñ for all 3   !. a) If 5L œ L , then 5LÐ3Ñ œ LÐ3Ñ for all 3. In particular, RK ÐLÑ © , RK ÐLÐ3Ñ Ñ =

3œ!

and so if O Ÿ K normalizes L , then O normalizes every LÐ3Ñ . b) The sequence of normal closures for 5L in K is 5L œ 5ÐLÐ=Ñ Ñ – 5ÐLÐ="Ñ Ñ – â – 5ÐLÐ!Ñ Ñ œ K and so =ÐLß KÑ œ =Ð5Lß KÑ.

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Proof. For part 2), we have 5ÐLÐ!Ñ Ñ œ Ð5LÑÐ!Ñ and if 5ÐLÐ3Ñ Ñ œ Ð5LÑÐ3Ñ , then 5ÐLÐ3"Ñ Ñ œ ncÐ5Lß 5ÐLÐ3Ñ ÑÑ œ ncÐ5Lß Ð5LÑÐ3Ñ Ñ œ Ð5LÑÐ3"Ñ

…

*Joins and Subnormality The question of whether the join of subnormal subgroups is subnormal is much more involved than the same question for intersection. Of course, if O is subnormal in K and if L is normal in K, then we may lift the sequence of normal closures of O in K O œ OÐ>Ñ – OÐ>"Ñ – â – OÐ!Ñ œ K

(4.29)

by L to get a series for the join LO . Thus, L ü Kß

O üüK

Ê

LO ü ü K

Actually, the process of lifting by L is the same as projecting the sequence (4.29) into the extension L ü K. If neither factor L or O is normal, we are tempted to project (4.29) into each extension LÐ3Ñ – LÐ3"Ñ and concatenate the resulting series. This almost works, the problem being that it produces a series from LÐ=Ñ ÐOÐ>Ñ ∩ LÐ="Ñ Ñ œ LO ∩ LÐ="Ñ to K, rather than from LO to K. So a slight modification is in order. Note that the unwanted LÐ="Ñ comes from the upper endpoint of the extension LÐ=Ñ – LÐ="Ñ . Thus, if we assume that O normalizes L , then O normalizes LÐ3Ñ for all 3 and so LÐ3Ñ O is a subgroup of K and LÐ3Ñ ü LÐ3"Ñ O . If we now project (4.29) into the extension LÐ3Ñ ü LÐ3"Ñ O , the result is a series [3 with lower endpoint LÐ3Ñ ÐOÐ>Ñ ∩ LÐ3"Ñ OÑ œ LÐ3Ñ O and upper endpoint LÐ3"Ñ O , that is, [3 À LÐ3Ñ O ü â ü LÐ3"Ñ O Moreover, since [Ð3"Ñ and [Ð3Ñ are contiguous, the concatenation [Ð=Ñ [Ð="Ñ â[Ð"Ñ is a series from LO to K and so LO ü ü K. Note also that each of the series [3 has length at most > and so =ÐLOß KÑ Ÿ =ÐLß KÑ=ÐOß KÑ

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Theorem 4.30 If Lß O ü ü K and if O normalizes L , then LO œ ØLß OÙ ü ü K and =ÐØLß OÙß KÑ Ÿ =ÐLß KÑ=ÐOß KÑ

…

*The Subnormal Join Property Theorem 4.30 implies the following useful characterization of the join question. Theorem 4.31 Let K be a group and let Lß O ü ü K. The following are equivalent: 1) ØLß OÙ ü ü K 2) L O ü ü K 3) ÒLß OÓ ü ü K. Proof. Recall from Theorem 3.40 that ÒLß OÓ ü LÒLß OÓ œ L O ü L O O œ ØLß OÙ Thus, 1) Ê 2) Ê 3). On the other hand, if 3) holds, then since L ü ü K and ÒLß OÓ ü ü K and L normalizes ÒLß OÓ, Theorem 4.30 implies that L O œ LÒLß OÓ ü ü K and so 2) holds. If 2) holds, then 1) holds since O normalizes L O .… Definition If a group K has the property that the join of any two subnormal subgroups is subnormal, then K is said to have the subnormal join property, or SJP.… Theorem 4.31 gives us one simple criterion for the SJP. If the commutator subgroup Kw of K has the property that all of its subgroups are subnormal, then ÒLß OÓ is subnormal in Kw and therefore also in K, for all Lß O Ÿ K and so Theorem 4.31 implies that K has the SJP. In particular, this occurs when Kw is abelian, that is, when K is metabelian. Theorem 4.32 If the commutator subgroup Kw of K has the property that all of its subgroups are subnormal, in particular, if K is metabelian, then K has the subnormal join property.… On a different line, if Kw has the ACC on subnormal subgroups (a finiteness condition), then K has the subnormal join property. Theorem 4.33 (Robinson) Let K be a group for which Kw has the ACC on subnormal subgroups. Then K has the subnormal join property.

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Proof. Let L and O be subnormal in K . The proof is by induction on = œ =ÐLß KÑ. If = Ÿ ", the result is clear, so assume that =   #. If + − K then + LÐ"Ñ œ ncÐL + ß KÑ œ ncÐLß KÑ œ LÐ"Ñ

and so L and L + are subnormal subgroups of LÐ"Ñ . But LÐ"Ñ also has the property that its commutator subgroup has the ACC on subnormal subgroups. w w w To see this, note that LÐ"Ñ « LÐ"Ñ ü K implies that LÐ"Ñ ü K and so subnÐLÐ"Ñ Ñ w is contained in subnÐK Ñ. Thus, since =ÐLß LÐ"Ñ Ñ œ =  ", the inductive hypothesis implies that ØLß L + Ù ü ü LÐ"Ñ and so ØLß L + Ù ü ü K. Moreover, this can be extended to more than one conjugate of L . For example, if , − K, then ØLß L + Ù, œ ØL , ß L ,+ Ù and so the join of ØLß L + Ù and ØLß L + Ù, is the subnormal subgroup ØLß L + ß L , Ù. Note also that we can replace L by any conjugate of L and so ØLß L +" ß á ß L +8 Ù ü ü K for all +" ß á ß +8 − K. Next, we show that ØÒLß +" Óß á ß ÒLß +8 ÓÙ ü ü K for all +" ß á ß +8 − K. First, note that ØLß ÒLß +ÓÙ œ ØLß L + Ù and so ØLß ÒLß +" Óß á ß ÒLß +< ÓÙ œ ØLß L +" ß á ß L +< Ù ü ü K But Theorem 3.37 implies that L normalizes each ÒLß +3 Ó and so ØÒLß +" Óß á ß ÒLß +< ÓÙ ü ØLß ÒLß +" Óß á ß ÒLß +< ÓÙ ü ü K Now we can complete the proof. The ACC on subnÐKw Ñ implies that there is a finite subset M œ Ö+" ß á ß +8 × © O for which ÒLß OÓ œ QM ³ ØÒLß +" Óß á ß ÒLß +8 ÓÙ for if not, then there is a strictly increasing sequence QM"  QM#  â of subnormal subgroups of Kw . Hence, ÒLß OÓ ü ü K.… We refer the reader to Robinson [26], page 389, for an example of a group that does not have the SJP.

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*The Generalized Subnormal Join Property It is not necessarily the case that a group with the SJP also has the property that the join of any family of subnormal subgroups of K is subnormal. This is called the generalized subnormal join property or GSJP. Theorem 4.34 (Wielandt [35], 1939) Let K be a group. 1) If K has the ACC on subnÐKÑ, then K has the GSJP. 2) If K has BCC on subnÐKÑ, in particular, if K is finite, then subnÐKÑ is a complete sublattice of subÐKÑ.… Proof. For part 1), since Kw also has the ACC on subnormal subgroups, K has the SJP. Hence, subnÐKÑ is closed under finite join and so Theorem 1.6 implies that K has the GSJP.… Robinson has provided the following characterization of the GSJP, whose proof uses ordinal numbers. Theorem 4.35 (Robinson) A group K has the generalized subnormal join property if and only if the union of every chain of subnormal subgroups is subnormal. Proof. If K has the GSJP, then the union of a chain of subnormal subgroups is the join of that chain and so is subnormal. For the converse, we first show that K has the SJP by induction on the smaller of the two subnormal indices. Let Lß O ü ü K and let = œ minÖ=ÐLß KÑß =ÐOß KÑ× If = Ÿ ", then ØLß OÙ ü ü K. Assume that =   # and that the result holds when the subnormal index is less than =. Well order the set O so that O œ Ö5α ± α  $ × where $ is an ordinal number and 5! œ ". Let P" œ ØL 5α ± α  " Ù be the join of the first " conjugates of L by elements of O . Then P" Ÿ LÐ"Ñ

and

P$ œ L O œ LÒLß OÓ

To show that P$ is subnormal in K, we use transfinite induction on " . If " œ !, then P" œ L ü ü K. For successor ordinals, if P" ü ü K, then P" and L 5" are contained in LÐ"Ñ and =ÐL 5" ß LÐ"Ñ Ñ œ =  ". Also, LÐ"Ñ has the property that the union of any chain of subnormal subgroups is subnormal. Hence, the induction hypothesis implies that

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135

P" " œ ØP" ß L 5" Ù ü ü LÐ"Ñ ü K Finally, if - is a limit ordinal, then

P- œ .ÖPα ± α  -×

is the union of a chain of subnormal subgroups, which is subnormal in K by hypothesis. Hence, the transfinite induction is complete and P$ ü ü K, whence ØLß OÙ ü ü K by Theorem 4.31. Thus, K has the SJP. Now, let f © subnÐKÑ. The join is not affected by including in f the join of every finite subset of f , so we may assume that f is closed under finite joins. For convenience, we refer to a family c © subnÐKÑ as good if c contains f and is closed under finite joins and chain joins. Then subnÐKÑ is good and the intersection f w of all good families is the smallest good family containing f . Let N œ 2f

and N w œ 2f w

Then N © N w . But if N § N w , then there is an W w − f w for which W w © y N and so if g œ ÖW − f w ± W © N × it follows that f © g § f w . But g is also good and so the minimality of f w implies that N œ N w . Finally, since f w is closed under chain join, Zorn's lemma implies that f w has a maximal member Q . If Q § N w , then there is an R − f w for which R © y Q and so Q ” R − f w , which contradicts the maximality of Q , whence Q œ N w and 2f œ 2f w œ Q − f w © subnÐKÑ

Thus, K has the GSJP.…

When All Subgroups Are Subnormal There are a variety of ways to characterize finite groups in which all subgroups are subnormal. (We will characterize groups in which all subgroups are normal in Theorem 5.20.) To explore this further, we need some additional terminology. By definition, the normalizer RK ÐLÑ of a subgroup L of K is the largest subgroup of K in which L is normal. Thus, a subgroup that is equal to its own normalizer is as “unnormal” as possible, since it is normal only in itself. Definition Let K be a group. 1) A subgroup L Ÿ K is self-normalizing if L œ RK ÐLÑ. 2) A group K has the normalizer condition if K has no proper selfnormalizing subgroups, that is, if L K

Ê

L  RK ÐLÑ

…

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Note that the term self-normalizing is a bit misleading, since every subgroup normalizes itself. (We would prefer the term unnormal.) A proper subnormal subgroup L cannot be self-normalizing, since the first step in a series shows that L is normal in some subgroup larger than itself. Hence, if all subgroups of K are subnormal, then K has the normalizer condition. The converse is also true if K is finite (or more generally, if K has the ACC on subgroups), since the proper series of normalizers L – RK ÐLÑ – RK ÐRK ÐLÑÑ – â must eventually reach K. Theorem 4.36 The following are equivalent for a finite group K: 1) Every subgroup of K is subnormal 2) K has the normalizer condition.… If K has the normalizer condition, then any maximal subgroup Q must be normal in K, for we have Q  RK ÐQ Ñ and the maximality of Q implies that RK ÐQ Ñ œ K. Thus, with respect to the following conditions on a finite group K: 1) 2) 3) 4)

Every subgroup of K is subnormal K has the normalizer condition Every maximal subgroup of K is normal KÎFÐKÑ is abelian

we have proved (see Theorem 4.22) that 1) Í 2) Ê 3) Í 4) For finite groups, it is our eventual goal to prove not only that these four conditions are equivalent, but that they are equivalent to several other conditions, one of which is that K satisfy a strong converse of Lagrange's theorem, namely, if 8 ± 9ÐKÑ, then K has a normal subgroup of order 8.

Chain Conditions Let us take a closer look at chain conditions for groups, beginning with the definition. Definition Let K be a group and let f be a family of subgroups of K. 1) A group K satisfies the ascending chain condition (ACC) on f if every ascending sequence L" Ÿ L# Ÿ â of subgroups in f must eventually be constant, that is, if there is an 8  !

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137

such that L85 œ L8 for all 5   !. In this case, we also say that f has the ACC. 2) A group K satisfies the descending chain condition (DCC) on f if every descending sequence L"   L#   â of subgroups in f must eventually be constant, that is, if there is an 8  ! such that L85 œ L8 for all 5   !. In this case, we also say that f has the DCC. 3) A group K satisfies both chain condition (BCC) on f if K has the ACC and the DCC on f . In this case, we also say that f has the BCC.… Theorem 1.5 implies that K has the BCC on f if and only if f has no infinite chains. Our main interest will center on the case where K is a group, R ü K and f is one of the following families: subÐR à KÑß

norÐR à KÑß

subnÐR à KÑ

of all subgroups, normal subgroups and subnormal subgroups, respectively, of K that contain R . The ACC and DCC are, in general, independent of each other, that is, all four combinations are possible. For example, a finite group has both chain conditions on subgroups. An infinite cyclic group (such as the integers) has the ACC on subgroups but not the DCC on subgroups. The :-quasicyclic group ™Ð:∞ Ñ has the DCC on subgroups but not the ACC. Finally, the group of rational numbers  has neither chain condition on subgroups. The chain conditions can be characterized as follows. Definition Let K be a group and let f © subÐKÑ. 1) K has the maximal condition on f if every nonempty subfamily of f contains a maximal element. 2) K has the minimal condition on f if every nonempty subfamily of f contains a minimal element.… Theorem 4.37 Let K be a group and let f © subÐKÑ. 1) K has the maximal condition on f if and only if K has the ACC on f . 2) K has the minimal condition on f if and only if K has the DCC on f . Proof. Suppose K satisfies the maximal condition on f and that L" Ÿ L# Ÿ â is an ascending sequence of members of f . Then the subgroups L5 have a maximal member L8 , which implies that L85 Ÿ L8 for all 5   !. Conversely,

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suppose K satisfies the ACC on f and let ` be a nonempty subfamily of f . If L" − ` is not maximal, then there is an L# − ` for which L"  L# . If L# is not maximal, then there is an L$ for which L"  L#  L$ . This must stop after a finite number of steps and so ` must have a maximal member. The proof of part 2) is analogous.… In certain cases, the ACC can be characterized in another important way. If f © subÐKÑ is closed under arbitrary intersections, then every subset \ of K is contained in a smallest element of f , called the f -closure of \ , which is Ø\Ùf œ ,ÖL − f ± \ © L×

Note that the set on the right is nonempty, since the assumption that f is closed under arbitrary intersections implies that f contains the empty intersection, which is K itself. We say that Ø\Ùf is f-generated by \ and if \ is a finite set, we say that Ø\Ùf is finitely f -generated by \ . Theorem 4.38 Let K be a group and let f © subÐKÑ be closed under arbitrary intersections and closed under unions of ascending sequences. Then K has the ACC on f if and only if every L − f is finitely f -generated. Proof. Suppose K satisfies the ACC on f and let L − f . If L is not finitely f generated, then for any 2" − L , we have Ø2" Ùf  L Hence, there is an 2# − L Ï Ø2" Ùf for which Ø2" Ùf  Ø2" ß 2# Ùf  L We can continue to choose elements to produce an infinite strictly ascending sequence, in contradiction to the ACC on f . Hence, L is finitely f -generated. Conversely, suppose every element of f is finitely f -generated and let L" Ÿ L# Ÿ â

be an ascending sequence of subgroups in f . Then L œ - L5 − f and so L œ Ø\Ùf for some finite set \ . Hence, there is an index 7 for which \ © L7 and so L œ L7 . But then L75 œ L for all 5   !.… Note that in the preceeding theorem, f can be subÐKÑ, norÐKÑ, the family of all characteristic subgroups of K or the family of all fully-invariant subgroups of K. We next describe how the chain conditions are inherited. Theorem 4.39 Let K be a group and let R ü K. Let Y ÐR à KÑ be one of the following families of subgroups of K:

Homomorphisms, Chain Conditions and Subnormality

subÐR à KÑß

norÐR à KÑß

139

subnÐR à KÑ

and let Y ÐKÑ œ Y ÐÖ"×ß KÑ. Write Y − ACC to denote the fact that Y has the ACC, and similarly for the DCC. 1) (Quotients) Y ÐR à KÑ − ACC

Í

Y ÐKÎR Ñ − ACC

2) (Extension) Y ÐR Ñß Y ÐKÎR Ñ − ACC

Ê

Y ÐKÑ − ACC

3) (Direct products) Y ÐK" Ñß Y ÐK# Ñ − ACC

Í

Y ÐK" } K# Ñ − ACC

Similar statements hold for the DCC in place of the ACC. Proof. Part 1) follows from the correspondence theorem. For part 2), let K" Ÿ K# Ÿ â be an ascending chain in Y ÐKÑ. The sequences K" ∩ R Ÿ K # ∩ R Ÿ â and K" R Ÿ K # R Ÿ â are ascending chains in Y ÐR Ñ and Y ÐR à KÑ, respectively. Since part 1) implies that Y ÐR à KÑ has the ACC, each sequence is eventually constant and so there is an index 7 for which K73 ∩ R œ K7 ∩ R

and K73 R œ K7 R

for all 3   !. Hence, Theorem 2.18 implies that K73 œ K7 for all 3   ! and so Y ÐKÑ − ACC. A similar argument holds for the DCC. For part 3), let Y ÐK" Ñ, Y ÐK# Ñ − ACC and let T œ K" } K# ß

R" œ K" } Ö"×ß

R# œ Ö"× { K#

Then Y ÐK3 Ñ − ACC implies that Y ÐR3 Ñ − ACC for 3 œ "ß #. Also, T ÎR" ¸ R# and so Y ÐT ÎR" Ñ − ACC. Hence, Y ÐT Ñ − ACC by the extension property. Conversely, if Y ÐT Ñ − ACC, then Y ÐR3 Ñ © Y ÐT Ñ implies that Y ÐR3 Ñ − ACC and so Y ÐK3 Ñ − ACC for 3 œ "ß #.…

Chain Conditions and Homomorphisms The presence of a chain condition can have a significant impact on homomorphisms. For example, if K has the ACC on normal subgroups, then any surjective endomomorphism 5À K Ä K is also injective. To see this, consider the kernel sequence of 5:

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kerÐ5Ñ Ÿ kerÐ5# Ñ Ÿ kerÐ5$ Ñ Ÿ â Since this must eventually be constant, we have kerÐ58 Ñ œ kerÐ585 Ñ for all 5   !. Now, if + − kerÐ5Ñ, then + œ 58 , for some , − K and so " œ 5 + œ 5 8" , and so , − kerÐ58" Ñ œ kerÐ58 Ñ, whence + œ 58 , œ ". Thus, kerÐ5Ñ is trivial. Let us refer to the subgroups Ö5 Kß 5# Kß 5$ Ká × as the higher images of K and the sequence K   5K   5 # K   â as the image sequence of 5 . If K has the DCC on all subgroups, then any injective endomomorphism 5À K Ä K is also surjective, since the image sequence of 5 must eventually be constant and so 5 8 K œ 5 8" K for some 8. Hence, the injectivity of 5 implies that K œ 5K , whence 5 is surjective. Of course, if K has the DCC on normal subgroups only, then we can draw the same conclusion provided that 5 has normal higher images. Theorem 4.40 Let K be a group and let 5 − EndÐKÑ. 1) If K has the ACC on normal subgroups, then 5 surjective

5 injective

Ê

2) If K has the DCC on all subgroups or if K has the DCC on normal subgroups and 5 has normal higher images, then 5 injective

Ê

5 surjective

…

Note that 5 has normal higher images if 5 preserves normality in general. Definition Let K be a group. A homomorphism 5À K Ä L is normality preserving if R üK

Ê

5R ü L

…

The composition of two normality-preserving homomorphisms is normality preserving. For endomorphisms, a stronger condition than that of preserving normality is the following. Definition An endomorphism 5À K Ä K of a group K is normal if 5 commutes with all inner automorphisms #1 of K, that is, for any + − K, 5Ð+1 Ñ œ Ð5+Ñ1 for all 1 − K.…

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Of course, the composition of normal maps is normal. Also, it is easy to see that a normal endomorphism is normality preserving. One of the advantages of normal maps over other normality-preserving maps is that if 5À K Ä K is normal and if L ü K is 5-invariant, then 5lL À L Ä L is also normal.

Fitting's Lemma If K has both chain conditions on normal subgroups and if 5 − EndÐKÑ has normal higher images, then both the kernel and image sequences of 5 must eventually be constant and so there is an 7  ! for which O œ kerÐ57 Ñ œ kerÐ573 Ñ

and L œ 57 K œ 573 K

for all 3   !. There is much that can be said about the subgroups L and O . First, 5L œ L and 5O Ÿ O and so L and O are 5 -invariant. Also, if + − L ∩ O , then + œ 57 , and 57 + œ ". Hence, 5#7 , œ " and so , − kerÐ5#7 Ñ œ kerÐ57 Ñ, whence + œ ". Thus, L ∩ O œ Ö"×. To show that K œ L  O , if + − K, then there is a , − K for which 57 + œ 5#7 , and so +Ð57 Ð, " ÑÑ − kerÐ57 Ñ, whence + œ Ò+57 Ð, " ÑÓ57 Ð,Ñ − LO Thus, K œ imÐ57 Ñ  kerÐ57 Ñ Finally, since 57 ÐOÑ œ Ö"×, the map 5lO is nilpotent and since 5ÐLÑ œ L , the map 5lL is surjective. We have just proved the important Fitting's lemma, which we can make a bit more general than the previous argument. Theorem 4.41 (Fitting's lemma, 1934) Let K be a group with the BCC on normal subgroups and let 5 − EndÐKÑ have normal higher images. 1) There is an 7  ! for which K œ imÐ57 Ñ  kerÐ57 Ñ where a) L œ imÐ57 Ñ and O œ kerÐ57 Ñ are 5 -invariant b) 5lL is surjective and 5lO is nilpotent. 2) In particular, if K is indecomposable, then 5 is either nilpotent or an automorphism of K.…

Automorphisms of Cyclic Groups Let us examine the automorphism group of the cyclic groups. It is clear that an automorphism of a cyclic group is completely determined by its value on a generator and that this value also generates the group.

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The only generators of an infinite cyclic group G∞ Ð+Ñ are + and +" and so AutÐG∞ Ñ œ Ö+ß 7 × where 7 B œ B" for all B − G∞ . Now suppose that G8 œ Ø+Ù is cyclic of order 8. If 7 − AutÐG8 Ñ, then 7 + œ +5 for some " Ÿ 5  8. But 9Ð+Ñ œ 9Ð+5 Ñ and so we must have 5 − ™8‡ . Moreover, this condition uniquely determines an automorphism 75 defined by 75 Ð+3 Ñ œ +53 for ! Ÿ 3  8. Hence, there is precisely one automorphism 75 for each 5 − ™‡8 . Moreover, the map 5À ™‡8 Ä AutÐG8 Ñ defined by 55 œ 75 is an isomorphism, since it is clear that 5 is bijective and if 4ß 5 − ™‡8 and 45 œ ;8  < with < − ™‡8 , then 7< Ð+3 Ñ œ +3< œ +345 œ 75 Ð+34 Ñ œ 75 74 Ð+3 Ñ Thus, AutÐG8 Ñ ¸ ™‡8 . Theorem 4.42 For the cyclic groups, we have AutÐG∞ Ð+ÑÑ œ Ö+ß 7 À + È +" × and AutÐG8 Ð+ÑÑ œ Ö75 ± 5 − ™‡8 × ¸ ™‡8 where 75 is defined by 75 Ð+Ñ œ +5 . In particular, the automorphism group of a cyclic group is abelian.…

A Closer Look at ™‡8 The previous theorem prompts us to take a closer look at the groups ™‡8 . /7 Theorem 4.43 If 8 œ :"/" â:7 where the :3 are distinct primes and /3   ", then

™‡8 ¸ ™:‡/" } â } ™:‡/5 "

5

Moreover, ™‡8 is cyclic if and only if 8 œ #ß %ß :/ or #:/ where : is an odd prime. Proof. Let <3 œ :3/3 . For the direct product decomposition, consider the map 5À ™‡8 Ä } ™‡<3 defined by 5Ð?Ñ œ Ð? mod <" ß á ß ? mod <7 Ñ This map is a group homomorphism and if 5? œ Ð"ß á ß "Ñ, then ? ´ " mod <3 , that is, <3 ± Ð?  "Ñ for all 3. Since the <3 's are pairwise relatively prime, it

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143

follows that 8 ± Ð?  "Ñ, that is, ? œ " in ™‡8 . Thus, 5 is a monomorphism and since the domain and the range of 5 have the same size (see Theorem 2.30), 5 is an isomorphism. Now, Theorem 2.33 implies that ™‡8 is cyclic if and only if each factor ™‡:/3 is 3

cyclic and the orders 9Й‡:/3 Ñ œ :3/3 " Ð:3  "Ñ 3

are relatively prime. So let us take a look at ™‡:/ for : prime. Let :  #. To see that ™‡:/ is cyclic, first note that if / œ ", then ™‡: is cyclic since ™: is a field. Assume that /  ". It is sufficient to find elements in ™‡:/ of the relatively prime orders :  " and :/" . To find an element of order :  ", let + − ™‡:/ have order :5 7, where 7 ± :  ". Then +:

5

7

œ " mod :/" Ð:  "Ñ

and /  " implies that +:

5

7

œ " mod :

But if + is chosen so that Ð+ß :Ñ œ ", then Fermat's little theorem implies that 5 +: ´ + mod : and so +7 ´ " mod :, whence 7 œ :  ". Thus, 5 9Ð+Ñ œ :5 Ð:  "Ñ and so 9Ð+: Ñ œ :  ", as desired. Moreover, since :  #, the expression " œ "  : is in :-standard form and so Theorem 1.18 implies that Ð"  :Ñ:

/"

œ "  A:/

where : y± A. Hence, "  : has order :/" in ™:‡/ . Now consider the case : œ #. It is easy to see that ™‡# and ™‡% are cyclic. For /   $, the elements of ™‡#/ are odd integers. For "  #+ − ™‡#/ , it is easy to see by induction that 5

Ð"  #+Ñ# œ "  #5# B5 In particular, /#

Ð"  #+Ñ#

œ "  #/ B/# ´ " mod #/

which implies that ™‡#/ has exponent #/# and so cannot be cyclic.

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In summary, we can say that ™‡:/ is cyclic if and only if :  # or :/ œ # or :/ œ %Þ We can now piece together our facts. As mentioned earlier, ™‡8 is cyclic if and only if each factor ™‡:/3 is cyclic and the orders 3

9Й‡:/3 Ñ œ :3/3 " Ð:3  "Ñ 3

:3/3 " Ð:3

are relatively prime. Since  "Ñ is even unless :3 œ # and /3 œ ", there can be at most one factor involving an odd prime or ™‡% . Thus, ™8‡ is cyclic if and only if 8 œ #ß %ß :/ or #:/ where :  # is prime.…

Exercises 1.

Let 5À K Ä L be a group homomorphism. a) Prove that if W Ÿ K, then 5W Ÿ L . b) Prove that if 5 is surjective and W ü K , then 5W ü L . c) Prove that if X Ÿ L , then 5" X Ÿ K . d) Prove that if X ü L , then 5" X ü K. 2. Let K and O be groups and let L Ÿ KÞ Show that it is not always possible to extend a homomorphism 5À L Ä O to K. 3. Show that the :-quasicyclic group ™Ð:∞ Ñ is isomorphic to the subgroup of all complex :8 th roots of unity. 4. Show that if K has a unique maximal subgroup Q , then KÎQ is cyclic of prime order. 5. Let K be a finite group with normal subgroups L and O . If KÎL ¸ KÎO , does it follow that L ¸ O ? 6. a) Find a property of groups that is inherited by quotient groups but not by subgroups. b) Find a property of groups that is inherited by subgroups but not by quotient groups. 7. Show that a group K is abelian if and only if the map + È +" is an automorphism of K. 8. Let K be a group. a) Determine all homomorphisms 5À G7 Ð+Ñ Ä K. b) Determine all homomorphisms 5À G∞ Ð+Ñ Ä K. 9. Are all normality-preserving homomorphisms normal? Hint: Use the fact that W$ is simple. 10. Let K be a group and let R be a normal cyclic subgroup of K. Prove that Kw Ÿ GK ÐR Ñ. 11. Let K be a group. Prove that Kw is central if and only if InnÐKÑ is abelian. 12Þ An endomorphism 5 − EndÐKÑ is central if +" 5+ − ^ÐKÑ for all + − K. This is equivalent to 5+^ÐKÑ œ +^ÐKÑ, that is, 5 acts like the identity on KÎ^ÐKÑ. Prove that a) A normal surjective endomorphism is central. b) A central endomorphism is normal.

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13. An abelian group E (written additively) is divisible if for any + − E and any positive integer 8, there is a , − E for which 8, œ +. Prove that a characteristically simple abelian group E is divisible. 14. Let K be a group and let + − K. Is the map 5À K Ä K defined by 5, œ Ò,ß +Ó an endomorphism of K? If not, under what conditions on K is 5 an endomorphism? 15. Let K be a group of order :# where : is a prime. a) Prove that if K is abelian, then K is either cyclic or isomorphic to the direct product of two cyclic groups of order :. b) Show that the distinct sets Ö+K × of conjugates of elements in K form a partition of K. Show that ^ÐKÑ Á Ö"×. Show that K must be abelian. 16. Prove that W$ ¸ H' , where W$ is the symmetric group of order ' and H' is the dihedral group of order '. 17. Prove that if K has a periodic subgroup L of finite index, then K is periodic. (In loose terms, if “most” of the elements of K have finite order, then all elements of K have finite order.) 18. Let K be a finite group. Let O ü K . A subgroup L of K is called a supplement of O if K œ LO . Let L be a minimal supplement of O . Prove that L ∩ O Ÿ FÐLÑ. 19. Let Y be a family of groups with the following properties: 1) If K − Y and L ¸ K, then L − Y . 2) If K − Y and R ü K, then KÎR − Y . 3) If R ü K and if R − Y and KÎR − Y , then K − Y . Prove that if R − Y and O − Y are subgroups of K with O ü K, then RO − Y. 20. If K is a finite group and 5 is an automorphism of K that fixes only the identity element, that is, 1 Á " implies 51 Á 1, show that K œ Ö1" 51 ± 1 − K× 21. Prove that if L ü ü K and O ü ü K, then it does not necessarily follow that the set product LO is a subgroup of K. Hint: Look at the dihedral group H) . 22. Let K be a finite group. Suppose that E ü K is minimal among all normal subgroups of K and that E is abelian. Prove that E is an elementary abelian group. 23. Show that a subgroup L of a group K is characteristic if and only if 5L œ L for all automorphisms 5 − AutÐKÑ. 24. Find an example of a normal subgroup L of a group K for which L is not characteristic in K. 25. Let K be a group. Prove the following: a) The property of being characteristic is transitive: If E « F and F « G , then E « G . b) If E « F and F ü G , then E ü G . 26. Show that ^ÐKÑ « K.

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27. Let K be a finite group and let L ü K. Show that if ÐkL kß ÐK À LÑÑ œ ", then L « K. 28. Let K be a group and let L Ÿ K. Prove that L « K implies ÒLß KÓ « K . 29. Let K be a finitely-generated group and let L Ÿ K with ÐK À LÑ  ∞. Show that there is a subgroup O Ÿ L that is characteristic in K and has finite index in K. 30. Let K be a group and let R – K. Let L  K . Show that R Ÿ FÐLÑ implies R Ÿ FÐKÑ. 31. Let K be a group. Let Kw be the commutator subgroup of K. Let Kww be the commutator subgroup of Kw . In general, we can continue to take commutator subgroups and KÐ8Ñ œ ÐKÐ8"Ñ Ñw is called the 8th commutator subgroup of K. Prove that KÐ8Ñ « K. 32. Let K be a group. Prove that if InnÐKÑ is cyclic, then K is abelian. 33. a) Show that the commutator subgroup of a group is fully invariant. b) Show that the center of a group need not be fully invariant. Hint: consider G# Ð+Ñ } W$ . /7 34. Let K be a finite abelian group of order 8 œ :"/" â:7 where the :3 's are distinct primes. For each prime :3 , let KÐ:3 Ñ œ Ö+ − K ± + is a :3 -element× a) Show that KÐ:3 Ñ « K. b) Show that

K œ 2KÐ:3 Ñ

and KÐ:3Ñ ∩ 24Á3 KÐ:3Ñ œ Ö"×

35. (Chinese remainder theorem) Let 7" ß á ß 75 be pairwise relatively prime integers greater than ". Let 0" ß á ß 08 be integers. Prove that the system of congruences B ´ 0" mod 7" ã B ´ 08 mod 75 has a unique solution modulo the product 7" â75 . : Hint: Use ™73 . 36. Let K be a group of order :8 where : is a prime. Show that a subgroup L of index : must be normal in K. Hint: Consider the map -À K Ä WKÎL , where WKÎL is the symmetric group on KÎL , defined by -Ð1ÑÐ+LÑ œ 1+L. 37. Find the subgroup lattice of U. Which subgroups are normal? Is U abelian? What is the center of U? What is Uw ? 38. Prove that the quasicyclic groups ™Ð:∞ Ñ are the only infinite groups with the property that their proper subgroups consist entirely of a single ascending chain Ö"×  W"  W#  â

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147

39. Let K be a group and let 8  " be an integer. a) When is the 8th power map 08 À K Ä K defined by 08 Ð+Ñ œ +8 a homomorphism? Must K be abelian? b) When is the 8th power map a homomorphism for all 8? c) Let K8 œ Ö+8 ± + − K× and let KÐ8Ñ œ Ö+ − K ± +8 œ "×. Show that both of these sets are normal subgroups of K. What is the relationship between these subgroups? 40. Find the automorphism group AutÐZ Ñ of the 4-group Z . 41. Let 5À K q» O be an epimorphism. Show that if L ü ü K, then 5L ü ü O and =Ð5 Lß OÑ Ÿ =ÐLß KÑ. 42. Prove that the subgroup Ø5Ù of the dihedral group 8

H#8" œ Ø5ß 3 ± 5# œ " œ 3# Ù has subnormal index 8. 43. If Y œ ÖL3 ± 3 − M× is a family of subnormal subgroups of a group K and if there is an integer = for which =ÐL3 ß KÑ Ÿ = for all 3 − M , show that +Y is subnormal in K. 44. Prove that any finitely-generated metabelian group has the generalized SJP. 45. A group K is Hopfian if K is not isomorphic to any proper quotient group of itself. A group K is co-Hopfian if K is not isomorphic to any proper subgroup of itself. a) Prove that K is Hopfian if and only if every endomorphism of K is an automorphism of K. b) Prove that K is co-Hopfian if and only if every monomorphism of K is an automorphism. c) Show that  is both Hopfian and co-Hopfian. Hint: To show that  is Hopfian, show that ÎL is not torsion free unless L œ Ö"×. To show that  is co-Hopfian, show that any proper subgroup L of  is not divisible. An abelian group K is divisible if + − K and 8 − ™ imply that there is a , − K for which + œ 8, . d) Show that ™ is Hopfian but not co-Hopfian. e) Show that the quasicyclic group ™Ð:∞ Ñ is co-Hopfian but not Hopfian. f) Show that the additive group of all polynomials in infinitely many variables is neither Hopfian nor co-Hopfian. 46. Find two nonisomorphic groups with isomorphic automorphism groups. 47. Prove that the multiplicative group  of positive rational numbers is isomorphic to the additive group ™ÒBÓ of polynomials over the integers. Hint: Use the fundamental theorem of arithmetic. 48. Let K be a group. Show that if K is centerless, then GAutÐKÑ ÐInnÐKÑÑ œ Ö+× 49. Show that AutÐW$ Ñ œ InnÐW$ Ñ ¸ W$ . 50. Prove that if K is a group in which every nonidentity element is an involution, then K has a nontrivial automorphism.

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51. Consider a series for K Ö"× œ K! ü K" ü K# ü â ü K8 œ K for which each factor group K3" ÎK3 is abelian. Show that if L is a subgroup of K, then there is a series of subgroups Ö"× œ L! ü L" ü L# ü â ü L7 œ L of L whose factor groups are also abelian. 52. (Robinson) Let R ü K and suppose that R has the GSJP and KÎR has the ACC on subnormal subgroups. Show that K also has the GSJP. Hint: Apply Theorem 4.35. Let V œ ÖL3 ± 3 − M× be an arbitrary chain in subnÐKÑ and let Y œ -V. To show that Y ü ü K, consider the chains VR œ ÖL3 R ± 3 − M× and VR ÎR œ ÖL3 R ÎR ± 3 − M× in K and KÎR , respectively. Let L" R ÎR be a maximal member of VR ÎR . Show that Y œ L" ÐY ∩ R Ñ.

Chapter 5

Direct and Semidirect Products

In this chapter, we will explore the issue of the decomposition of groups into “products” of subgroups. To say simply that a group K can be decomposed into a set product of two proper subgroups K œ LO leaves something to be desired. The first problem is that the representation of an element of K as a product 25 for 2 − L and 5 − O need not be unique. The second problem is that, in general, we have no information about how the elements of L and O interact, for example, what is 52 as a member of LO ? The first problem is easily addressed. The second is not as simple.

Complements and Essentially Disjoint Products Uniqueness in a set product decomposition is easily characterized. Theorem 5.1 Let Lß O Ÿ K. 1) Every element + − LO has a unique representation as a product + œ 25 for 2 − L and 5 − O if and only if L and O are essentially disjoint. 2) Every element + − K can be uniquely represented as an element of LO if and only if K œ L ì O , that is, if and only if K œ LO

and L ∩ O œ Ö"×

In this case, O is said to be a complement of L in K. A subgroup L of K is complemented if it has a complement in K.… Note that since K œ LO implies K œ OL , the concept of complement is symmetric in L and O . Also, if K œ L ì O , then kKk œ kL ì O k œ kL kkO k

as cardinal numbers.

S. Roman, Fundamentals of Group Theory: An Advanced Approach, DOI 10.1007/978-0-8176-8301-6_5, © Springer Science+Business Media, LLC 2012

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We pause for a small clarification of terminology. In the more specialized literature of group theory, devoted to the study of the subgroup lattice subÐKÑ of a group K, such as appears in the book Subgroup Lattices of Groups, by Schmidt [30], the term complement of L − subÐKÑ refers to a subgroup O for which K œL ”O

and L ∩ O œ Ö"×

This terminology is consistent with the terminology of lattice theory. Indeed, Schmidt uses the term permutable complement for the concept given in our definition. However, our definition follows the trend in general treatments of group theory (including most textbooks). Even for normal subgroups, complements need not exist. For example, no nontrivial proper subgroup of ™ has a complement. Moreover, when complements do exist, they need not be unique; for example, in W$ every subgroup of order # is a complement of the alternating subgroup E$ and so, for example, W$ œ E$ ì ØÐ" #ÑÙ œ E$ ì ØÐ" $ÑÙ are two essentially disjoint product representations of W$ . On the other hand, any complement O of a normal subgroup R is isomorphic to the quotient KÎR , for we have K R ìO O O œ ¸ œ ¸O R R R ∩O Ö"× Theorem 5.2 Let K be a group. If a normal subgroup R of K is complemented, then all complements of R are isomorphic to KÎR and hence to each other.…

Complements and Transversals Another way to characterize complements is through the notion of a transversal. Let us recall the following definition. Definition Let L Ÿ K. 1) A set consisting of exactly one element from each left coset in KÎL is called a left transversal for L in K (or for KÎL ). 2) A set consisting of exactly one element from each right coset in LÏK is called a right transversal for L in K (or for LÏK).… It is of interest to know which subgroups of K are left transversals for L . The simple answer is that these are precisely the complements of L . Theorem 5.3 Let L Ÿ K. The following are equivalent for a subgroup O Ÿ K. 1) O is a complement of L in K.

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2) O is a left transversal for L in K. 3) O is a right transversal for L in K. Proof. Suppose that O is a complement of L in K. If 5" L œ 5# L , then 5#" 5" − O ∩ L œ Ö"× and so 5" œ 5# . Also, since K œ OL , every + − K has the form + œ 52 − 5L for some 5 − O . Thus, the cosets 5L for 5 − O form a left transversal for L in K . Conversely, suppose that O Ÿ K is a left transversal for KÎL . Since L contains a single member of O , it follows that L ∩ O œ Ö"×. Also, since the cosets KÎL partition K , any + − K has the form + œ 52, for some 5 − O and 2 − L and so K œ OL . Thus, O is a complement of L in K. A similar argument can be made for the right cosets.…

Product Decompositions If K œ L ì O , then every element of K has a unique representation as a product 25 with 2 − L and 5 − O . The problem, however, is that we have no information about how the elements of L and O interact. Without a commutativity rule that expresses a product 52 in the form 2w 5 w , where 2ß 2w − L and 5ß 5 w − O , we have no way to simply a product of the form Ð2" 5" ÑÐ2# 5# Ñ The simplest commutativity rule, that is, elementwise commutativity 25 œ 52 holds if and only if both factors L and O are normal in K and this essentially reflects the fact that there is no interaction between L and O . Weaker forms of decomposition come by weakening the requirement that both factors be normal. Here are the relevant definitions in one place for comparison purposes. Definition Let K be a group and let Lß O Ÿ K. 1) If K œL ìO then K is called the essentially disjoint product of L and O . In this case, L is called a complement of O in K . 2) If K œ L ì Oß

LüK

then K is called the semidirect product of L with O . In this case, L is called a normal complement of O in K. The semidirect product is denoted by K œL zO 3) If K œ L ì Oß

L ü Kß O ü K

then K is called the direct product of L and O . In this case, L is called a direct complement of O in K (and vice versa). The direct product is

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Any of these products is nontrivial if both factors are proper.… The direct product decomposition of a group is a very strong form of decomposition and so is rather specialized. However, the semidirect product is one of the most useful constructions in group theory. For example, for 8 Á # mod %, the dihedral groups H#8 have no direct product decompositions, but they do have a semidirect product decomposition H#8 œ Ø3Ù z Ø5Ù Nevertheless, all three types of product decompositions are special. To illustrate the point, an infinite cyclic group has no nontrivial essentially disjoint product decompositions at all.

Direct Sums and Direct Products External Direct Sums and Products We have already defined the external direct product of a finite number of groups. The generalization of this product to arbitrary families of groups leads to two important variations. Intuitively speaking, if , is an arbitrary (finite or infinite) cardinal number, we can consider the set of all ordered “, -tuples” as well as the set of all ordered ,-tuples that have only a finite number of nonzero coordinates. The notion of an ordered ,-tuple is generally described by a function. Definition Let Y œ ÖK3 ± 3 − M× be a family of groups. 1) The external direct product of the family Y is the group

}Y œ } K3 œ š0 À M Ä .K3 ¹ 0 Ð3Ñ − K3 ›

of all functions 0 from the index set M to the union of Y for which the 3th coordinate 0 Ð3Ñ of 0 belongs to K3 for all 3. The group operation is componentwise product: Ð0 1ÑÐ3Ñ œ 0 Ð3Ñ1Ð3Ñ The support of 0 − } K3 is the set suppÐ0 Ñ œ Ö3 − M ± 0 Ð3Ñ Á "× 2) The external direct sum of the family Y is the group

{Y œ { K3 œ e0 − }Y ± 0 has finite supportf

also under componentwise product.…

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153

Of course, when Y is a finite family, the direct product and direct sum coincide. Note that authors vary on their use of the terms direct sum and direct product. For example, some authors reserve the term sum for abelian groups and some authors use the term cartesian product for direct product.

Internal Direct Products For convenience, we repeat the definition of the internal direct product. Definition A group K is the (internal) direct sum or (internal) direct product of a family Y œ ÖL3 ± 3 − M× of normal subgroups if Y is strongly disjoint and K œ 1Y . We denote the internal direct product of Y by or

 L3

Y

or when Y œ ÖL" ß á ß L8 × is a finite family, L"  â  L8 Each factor L3 is called a direct summand or direct factor of K. We denote the family of all direct summands of K ,C Wf ÐKÑ.… We should mention that the notation for direct sums and products varies considerably among authors. For example, some authors use the notation K œ L ‚ O for both the internal and external direct sum (as well as the cartesian product), justifying this on the grounds that the two types of direct sums are isomorphic. While this may be reasonable, in an effort to avoid any ambiguity, we have adopted the following notations: 1) Set product LO 2) Essentially disjoint set product L ìO 3) Cartesian product of sets L ‚O 4) External direct product L }O

and

} L3

L {O

and

{ L3

and

 L3

5) External direct sum

6) Internal direct product (sum) LO

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7) Semidirect product L zO Note that our notation for the internal direct product emphasizes the fact that the two subgroups are normal (a juxtaposition of the symbols — and – used to indicate normality) and is similar to the established notation z for the semidirect product. To see that internal and external direct sums are isomorphic, suppose that K œ { K3 . For each 5 , let L5 œ Ö0 − { K3 ± 0 Ð3Ñ œ " for 3 Á 5× We leave it as an exercise to show that L5 ü K and that K œ  L5 . On the other hand, if K œ  K3 , then the map 5À { K3 Ä  K3 defined by 50 œ $Ö0 Ð3Ñ ± 3 − Mß 0 Ð3Ñ Á "×

is an isomorphism. For this reason, many authors drop the adjectives “internal” and “external” when discussing direct sums.

The Universal Property of Direct Products and Direct Sums Direct products and direct sums can each be characterized by a universal property. Projection and Injection Maps If K œ } ÖL3 ± 3 − M×

or K œ { ÖL3 ± 3 − M×

is an external direct product or external direct sum, we define the 3th projection map 33 À K Ä L3 by 33 Ð0 Ñ œ 0 Ð3Ñ Note that 33 is an epimorphism and that an element 0 − K is uniquely determined by the values 33 Ð0 Ñ, which can be specified arbitrarily, as long as 33 Ð0 Ñ − L3 for all 3. The 3th injection map ,3 À L3 Ä K is defined by ,3 Ð2ÑÐ4Ñ œ œ

2 "

if 4 œ 3 otherwise

for all 2 − L3 . These maps are injective. Note also that + 34 ‰ , 3 œ œ ! For an internal direct sum

if 3 œ 4 otherwise

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155

K œ  ÖL3 ± 3 − M× we have already defined the projection 33 À K Ä L3 by setting 33 Ð+Ñ to be the 3th component of +. For the internal direct sum, the injection maps ,3 À L3 Ä K are just the inclusion maps, defined by ,3 Ð2Ñ œ 2 for all 2 − L3 . Universality of Direct Products Let ÖL3 ± 3 − M× be a family of groups and let T œ } ÖL3 × 3−M

with projection maps 33 . Then the ordered pair c œ T ß Ö33 ×3−M has a universal property that characterizes the direct product up to isomorphism.

G fj

fi ∃!τ

Hi

ui

U

uj

Hj

Figure 5.1 Definition Referring to Figure 5.1, let Y be the family of all ordered pairs Z œ Kß Ö03 À K Ä L3 ×3−M where K is a group and Ö03 À K Ä L3 ×3−M is a family of homomorphisms. We refer to K as the vertex of the pair Z . A pair h œ Y ß Ö?3 À Y Ä L3 ×3−M in Y is universal for Y if for any pair Z − Y , there is a unique homomorphism 7 À K Ä Y between the vertices, called the mediating morphism for Z , for which ?3 ‰ 7 œ 03 for all 3 − M .… Theorem 5.4 Let ÖL3 ± 3 − M× be a family of groups. 1) The pair c œ T ß Ö33 À T Ä L3 ×3−M

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where T is the direct product of the family ÖL3 × and 33 is the 3th projection map, is universal for Y . 2) If the pair ÐKß Ö03 ×3−M Ñ is also universal for Y , then the mediating morphism 7À K Ä T is an isomorphism and so K ¸ T . Proof. For part 1), for any pair Z − Y , there must exist a unique mediating morphism 7À K Ä T satisfying 33 ‰ 7 œ 03 But this is equivalent to 33 Ð7 +Ñ œ 03 Ð+Ñ for all + − K and this does uniquely define a homomorphism 7 . The proof of part 2) is also to the proof of Theorem 4.5 and we leave the details to the reader.… Universality of Direct Sums Let ÖL3 ± 3 − M× be a family of groups and let W œ { ÖL3 ± 3 − M× with injections ,3 . The ordered pair f œ ÐWß Ö,3 ×3−M Ñ also has a universal property that characterizes the direct sum up to isomorphism.

Hi

ui fi

U

Hj

uj

∃!τ

fj

G Figure 5.2 Definition Referring to Figure 5.2, let Y be the family of all ordered pairs Z œ Kß Ö03 À L3 Ä K×3−M where K is a group and Ö03 À L3 Ä K×3−M is a family of homomorphisms. We refer to K as the vertex of the pair. A pair h œ Y ß Ö?3 À L3 Ä Y ×3−M − Y is universal for Y if for any pair Z − Y , there is a unique homomorphism 7 À Y Ä K between the vertices, called the mediating morphism for Z , for which

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7 ‰ ?3 œ 03 for all 3 − M .… Theorem 5.5 Let ÖL3 ± 3 − M× be a family of groups. 1) The pair f œ Wß Ö,3 À L3 Ä W×3−M where W is the direct sum of the family ÖL3 × and ,3 is the 3 th injection map, is universal for Y . That is, for any pair Z œ Kß Ö03 À L3 Ä K×3−M where K is a group and Ö03 À L3 Ä K×3−M is a family of homomorphisms, there is a unique homomorphism 7 À W Ä K for which 7 ‰ ,3 œ 0 3 for all 3 − M , or equivalently, 7lL3 œ 03 for all 3 − M . In other words, a homomorphism 7 À W Ä O is uniquely determined by its restrictions 7 lL3 to the factors L3 , which may be any homomorphisms from L3 to K. 2) If the pair ÐKß Ö03 ×3−M Ñ is also universal for Y , then the mediating morphism 7 À W Ä K is an isomorphism and so K ¸ W . Proof. For part 1), for any pair ÐOß Ö53 À L3 Ä O×3−M Ñ in Y , there must be a unique mediating morphism 7À W Ä K satisfying 7 ‰ ,3 œ 0 3 But this specifies how 7 is defined on any element of W that has support of size " and therefore on any element of W that has finite support, that is, on all of W . We leave the details of this and the proof of part 2) to the reader.…

Cancellation in Direct Sums A group K is cancellable in direct sums (or cancellable for short) if E  K ¸ F  Lß

K¸L

Ê

E¸F

We follow a line similar to Hirshon [18] to prove that any finite group is cancellable in direct sums. On the other hand, Hirshon shows that even infinite cyclic groups are not cancellable in direct sums. Theorem 5.6 Any finite group K is cancellable in direct sums. Proof. It is sufficient to show that E  K œ F  Lß

K¸L

Ê

E¸F

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and we prove this by induction on 9ÐKÑ. If 9ÐKÑ œ ", the result is clear. Assume that any group of order less than 9ÐKÑ is cancellable and let [ œ E  K œ F  Lß

K¸L

First, we observe that if F ∩ K œ Ö"×, then [ œ F  K , for we have FK [ ¸K¸L ¸ F F and since these groups are finite, they have the same size and so they are equal, whence F  K œ [ . It follows that E and F are both direct complements of K in [ and so are isomorphic. A similar argument holds if E ∩ L œ Ö"×. Thus, we may assume that F ∩ K and E ∩ L are nontrivial. Then EK LF œ ÐE ∩ LÑ  ÐF ∩ KÑ ÐE ∩ LÑ  ÐF ∩ KÑ and Theorem 5.19 implies that E K L F } ¸ } E∩L F∩K E∩L F∩K Since L ¸ K, we have L E K K L F } } ¸ } } Ö"× E ∩ L F∩K Ö"× E ∩ L F∩K and Theorem 4.8 implies that L }E}K L }K}F ¸ Ö"× } ÐE ∩ LÑ } ÐF ∩ KÑ Ö"× } ÐE ∩ LÑ } ÐF ∩ KÑ Splitting this in a different way gives E L K F L K } } ¸ } } Ö"× E ∩ L F∩K Ö"× E ∩ L F∩K Since 9ÐKÎÐF ∩ KÑÑ  9ÐKÑ, the induction hypothesis implies that KÎÐF ∩ KÑ is cancellable. Similarly, LÎÐE ∩ LÑ is cancellable and so E ¸ F .…

The Classification of Finite Abelian Groups We are now in a position to solve the classification problem for finite abelian groups, that is, we can identify a system of distinct representatives for the isomorphism classes of finite abelian groups. Theorem 5.7 Let E be a finite abelian group. If ? − E has maximum order among all elements of E, then Q œ Ø?Ù has a direct complement, that is, there

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159

is a subgroup Z for which EœQ Z Proof. The proof is by induction on the order of E. If 9ÐEÑ œ ", the result is clear. Assume the result holds for all abelian groups of order less than that of E. We may also assume that Q  E. If we find a subgroup \ Ÿ E for which \ ∩ Q œ Ö!×, since then 9Ð?  \Ñ œ 9Ð?Ñ, the inductive hypothesis applied to the quotient KÎ\ gives E Q \ Z Q Z œ  œ \ \ \ \ for some Z Ÿ E. Hence, E œ Q  Z and since ÐQ  \Ñ ∩ Z œ \ , it follows that Q ∩ Z © Q ∩ \ œ Ö!×, whence E œ Q  Z . To this end, let \ be a minimal subgroup of E that is not contained in Q . If B − \ , then ØBÙ is not contained in Q and so \ œ ØBÙ. If 9ÐBÑ œ +, with + and , relatively prime and greater than ", then Ø+BÙ  ØBÙ and so Ø+BÙ Ÿ Q . Similarly, Ø,BÙ Ÿ Q and so B − Ø+BÙ  Ø,BÙ Ÿ Q , a contradiction. Hence, 9ÐBÑ œ : is prime and so \ ∩ Q œ Ö!×.… Thus, if E is a finite abelian group and if ?" has maximum order in E, then E œ Ø?" Ù  Z" for some Z" Ÿ E. If Z" Á Ö!× and ?# has maximum order in Z" , then E œ Ø?" Ù  Ø?# Ù  Z# where 9Ð?# Ñ ± 9Ð?" Ñ. Since E is finite, this process must eventually result in a cyclic decomposition E œ Ø?" Ù  â  Ø?8 Ù of E, where if α5 œ 9Ð?5 Ñ, then α8 ± α8" ± â ± α" Moreover, if α" has the prime factorization /

/

/

/

α" œ :""ß" â:7"ß7 then α5 ± α" implies that α5 œ :"5ß" â:75ß7 where /5ß3 Ÿ /5"ß3 for all 3 œ "ß á ß 7. Then Theorem 2.33 implies that Ø?5 Ù œ Ø@5ß" Ù  â  Ø@5ß75 Ù /

where 9Ð@5ß3 Ñ œ :3 5ß3 for all 3. A subgroup of order a power of a prime : is called

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a :-primary (or just primary) subgroup. It is customary when writing E as a direct sum of primary cyclic subgroups to collect the terms associated with each prime. Theorem 5.8 (The cyclic decomposition of a finite abelian group) Let E be a finite abelian group. 1) (Invariant factor decomposition) E is the direct sum of a finite number of nontrivial cyclic subgroups E œ Ø?" Ù  â  Ø?8 Ù where if α5 œ 9Ð?5 Ñ, then α8 ± α8" ± â ± α" The orders α3 are called the invariant factors of E. 2) (Primary cyclic decomposition) If /

/

α5 œ :"5ß" â:75ß7 where /5ß3 Ÿ /5"ß3 for all 3 œ "ß á ß 7, then E is the direct sum of primary cyclic subgroups: E œ ÒØ?"ß" Ù  â  Ø?"ß5" ÙÓ  â  ÒØ?7ß" Ù  â  Ø?7ß57 ÙÓ /

/

where 9Ð?3ß4 Ñ œ :3 3ß4 . The numbers :3 3ß4 are called the elementary divisors of E.… We note the following: 1) The product of the invariant factors is equal to the product of the elementary divisors and is the order of the group. 2) The invariant factor α5 is equal to the maximum order of the elements of the group Ø?" Ù  â  Ø?5 Ù In particular, the largest invariant factor α" is equal to the maximum order of the elements of E. 3) The multiset of invariant factors of E uniquely determine the multiset of elementary divisors of E and vice versa. In particular, the elementary divisors are determined by factoring the invariant factors and the invariant factors are determined by multiplying together appropriate elementary divisors.

Uniqueness Although the invariant factor decomposition and the primary cyclic decomposition are not unique, the multiset of invariant factors and the multiset of elementary divisors are uniquely determined by E. Actually, the proof is

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quite easy if we use the cancellation property of Theorem 5.6. Suppose that E œ Ø?" Ù  â  Ø?8 Ù where α5 œ 9Ð?5 Ñ and α8 ± α8" ± â ± α" and that E œ Ø@" Ù  â  Ø@7 Ù where "5 œ 9Ð@5 Ñ and "7 ± "7" ± â ± "" Then α" and "" are both equal to the maximum order of the elements of E and so α" œ "" . Hence Ø?" Ù ¸ Ø@" Ù and Theorem 5.6 implies that Ø?# Ù  â  Ø?8 Ù ¸ Ø@# Ù  â  Ø@7 Ù Now, α# and "# are equal to the maximum orders in the two isomorphic groups above and so α# œ "# and we may cancel again. Hence, 7 œ 8 and α5 œ "5 for all 5 . Theorem 5.9 (Uniqueness) Let E be a finite abelian group. The multiset of invariant factors (and elementary divisors) for E is uniquely determined by E.…

Properties of Direct Summands Let us explore some of the properties of direct summands. The following simple facts are worth explicit mention: 1) The direct summand property is transitive: L − Wf ÐOÑß

O − Wf ÐKÑ

Ê

L − Wf ÐKÑ

2) The property of being a direct summand in inherited by subgroups: L − Wf ÐKÑß

L ŸOŸK

Ê

L − Wf ÐOÑ

In fact, K œ L  Nß

L ŸOŸK

Ê

O œ L  ÐN ∩ OÑ

3) Normal subgroups of direct summands are normal in the group: R ü Lß

L − Wf ÐKÑ

Ê

R üK

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4) The property of being characteristic is preserved by intersection with a direct summand: R « Kß

L − Wf ÐKÑ

Ê

R ∩L «L

Good Order Let K be a group and suppose that K œ E  Ew œ F  F w If E © F , then it does not necessarily follow that Ew ª F w , as is easily seen in the Klein %-group, for example. For convenience, let us say that an equation of the form K œ E  Ew œ F  F w ß

E©F

is in good order if Ew ª F w . Theorem 5.10 Let K be a group. If K œ E  Ew œ F  F w ß

E©F

then we can replace either Ew or F w by another subgroup to get an equation in good order. Specifically, the following are in good order: K œ E  ÒF w  ÐEw ∩ FÑÓ œ F  F w and K œ E  Ew œ F  ÐEw ∩ EF w Ñ Proof. The first equation follows directly from Dedekind's law. For the second equation, F ∩ ÐEw ∩ EF w Ñ œ Ew ∩ ÐF ∩ EF w Ñ œ Ew ∩ EÐF ∩ F w Ñ œ Ew ∩ Ew œ Ö"× and FÐEw ∩ EF w Ñ œ FEÐEw ∩ EF w Ñ œ FÐEEw ∩ EF w Ñ œ FEF w œ K

…

Chain Conditions on Direct Summands and Remak Decompositions Direct summands are also special with respect to chain conditions. We have seen that for the families subÐKÑ and norÐKÑ, the two chain conditions (ACC and DCC) are independent. However, for the family WfÐKÑ of direct summands of K, the two chain conditions are equivalent.

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Theorem 5.11 A group K has the ACC on Wf ÐKÑ if and only if it has the DCC on WfÐKÑ. Proof. Suppose that K has the DCC on direct summands and let H" Ÿ H# Ÿ â be an ascending sequence in WfÐKÑ. Theorem 5.10 implies that there is a descending sequence I"   I#   â where K œ H3  I3 for all 3. This descending sequence must eventually become constant, at which time so does the original sequence of H3 's.… A group with the the BCC on direct summands can be decomposed into a finite direct sum of indecomposable factors. Definition If K is a group, then any decomposition K œ V"  â  V8 where each V5 is indecomposible is called a Remak decomposition of K.… A minimal direct summand is a minimal member of the family WfÐKÑ Ï Ö"× of all nontrivial direct summands of K. A direct summand is minimal if and only if it is indecomposable. Theorem 5.12 (Remak) If a group K has either (and therefore both) chain condition on direct summands, then K has a Remak decomposition K œ V"  â  V8 Proof. Let V" be a minimal direct summand of K. If V" œ K, then K is indecomposable and we are done. Otherwise, K œ V"  I" where I" Á Ö"× also has BCC on direct summands. Let V# be a minimal direct summand of I" , which is also a minimal direct summand of K. If V# œ I" , we are done. If not, then K œ V "  V #  I# Since the sequence V"  V "  V #  â is a strictly increasing sequence of direct summands of K, it must become constant and so this construction must terminate after a finite number of steps, resulting in a decomposition of K into a finite direct sum of minimal direct summands.…

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A Maximality Property We paraphrase Theorem 3.15. Theorem 5.13 Let Y œ ÖL3 ± 3 − M× be a nonempty family of normal subgroups of a group K. For any O ü K, there is a N © M that is maximal with respect to the property that the direct sum O  Œ  L4 4−N exists.… One consequence of Theorem 5.13 is the following. Theorem 5.14 Suppose that K is the join of a family Y œ ÖL3 ± 3 − M× of minimal normal subgroups. For any O ü K, there is a N © M for which K œ O  Œ  L4 4−N

and so norÐKÑ œ Wf ÐKÑ Proof. Let N © M be maximal with respect to the fact that the direct sum QN œ O  Œ  R4 4−N exists. If 3  N , then QN ∩ R3 ü K implies that QN ∩ R3 œ Ö"× or R3 Ÿ QN . But if QN ∩ R3 œ Ö"×, then the direct sum QN ∪Ö3× exists, contradicting the maximality of N . Hence, R3 Ÿ QN for all 3 − M and so K œ QN .…

xY-Groups It is natural to ask the following types of questions. Let k ÐKÑ be a class of subgroups of a group K, such as the class of all subgroups, all normal subgroups or all characteristic subgroups. 1) For which groups K is it true that all subgroups in k ÐKÑ are complemented? 2) For which groups K is it true that all subgroups in k ÐKÑ have normal complements? 3) For which groups K is it true that all subgroups in k ÐKÑ are direct summands? There has been much research done on these and related questions and it has become customary to make the following types of definitions in this regard.

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Definition 1) An aC-group is a group in which all subgroups are complemented. 2) An nC-group is a group in which all normal subgroups are complemented. 3) An aD-group is a group in which all subgroups are direct summands. 4) An nD-group is a group in which all normal subgroups are direct summands. 5) An aNC-group is a group in which all subgroups have normal complements.… Note that an nNC-group is the same as an nD-group. We will characterize aDgroups, nD-groups and aNC-groups. We will also describe (without proof) the characterization of aC-groups. The theory of the least restrictive of these conditions—the nC-groups—appears to be much more complicated and we refer the reader to Christensen [8] and [9] for more details.

aD-Groups Groups in which all subgroups are direct summands were characterized by Kertész in 1952. This is the strongest of the xY-conditions and is indeed very restrictive. Theorem 5.15 (aD-groups: Kertész [21], 1952) A group K is an aD-group if and only if it is the direct product of cyclic groups of prime order. Proof. If K is a direct sum of cyclic subgroups G3 of prime order, then since each G3 is minimal normal, Theorem 5.14 implies that subÐKÑ œ norÐKÑ œ Wf ÐKÑ and so K is an aD-group. For the converse, suppose that K is an aD-group. Then any subgroup of K is also an aD-group. However, the only cyclic aD-groups Ø+Ù are those of squarefree order (that is, order a product of distinct primes). For it is clear that Ø+Ù cannot have infinite order and if :# ± 9Ð+Ñ for any prime :, then Ø+Ù has an element B of order : and so Ø+Ù œ ØBÙ  O where : ± 9ÐOÑ. Hence, O has an element of order : as well, which is too many elements of order : for a cyclic group. Hence, every element of K has squarefree order. Now, consider the family Y of all cyclic subgroups of prime order in K. If + − K has order 9Ð+Ñ œ :" â:7 , where the factors are distinct primes, then Theorem 2.33 implies that there are subgroups Ø+3 Ù of order :3 for which Ø+Ù œ Ø+" Ù  â  Ø+7 Ù

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and so K œ 1Y . But since each member of Y is minimal normal, Theorem 5.14 implies that K is the direct sum of a subfamily of Y .…

nD-Groups The nD-condition is not as strong as the aD-condition, but is still very restrictive. One reason is that normality is a finitary condition, involving individual elements and so the condition WfÐKÑ œ norÐKÑ imparts finitary properties to the otherwise global condition of being a direct summand. In particular, the union of any ascending sequence of normal subgroups is normal and so in an nD-group, the union of any ascending sequence of direct summands is a direct summand. This condition implies a certain measure of finiteness for nD-groups. Specifically, if K is a nontrivial nD-group and H" Ÿ H# Ÿ â is an ascending chain of direct summands in R , then

K œ Š.H3 ‹  I œ .ÐH3  IÑ

for some I Ÿ K. Hence, if K œ ncÐWß KÑ is the normal closure of a finite subset W © K, then W © H8  I for some 8   " and so K œ H8  I , which implies that H83 œ H8 for all 3   !. Thus, K has the ACC on direct summands and so K has a Remak decomposition. Note also that any normal subgroup (direct summand) H of an nD-group K is an nD-group, since Wf ÐHÑ œ Wf ÐKÑ ∩ subÐHÑ œ norÐKÑ ∩ subÐHÑ œ norÐHÑ It follows that any nontrivial nD-group contains an indecomposable direct summand. We can now characterize nD-groups. Suppose that K is a nontrivial nD-group and consider the family Y œ ÖW3 ± 3 − M× of all indecomposable direct summands of K, along with the trivial subgroup. Theorem 5.13 implies that there is a subset N © M that is maximal with respect to the fact that the direct sum W œ  W4 4−N

exists in K and so the nD-condition implies that KœWL for some L Ÿ K. But if L is not trivial, then it contains an indecomposable direct summand of K, which contradicts the maximality of N . Hence, L is trivial and

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K œ  W4 4−N

is a direct sum of indecomposable subgroups. In particular, each W3 is minimal normal in K and so K is the direct sum of minimal normal subgroups. Conversely, if K is the direct sum of minimal normal subgroups, then Theorem 5.14 implies that norÐKÑ œ Wf ÐKÑ, that is, K is an nD-group. Theorem 5.16 The following are equivalent for a group K: 1) K is an nD-group 2) K is the direct sum of minimal normal subgroups 3) K is the direct sum of simple subgroups.…

aNC-Groups We next show that aNC-groups are the same as aD-groups, by showing that an aNC-group is abelian. Theorem 5.17 (aNC-groups: Weigold [34], 1960) A group is an aNC-group if and only if it is an aD-group, that is, if and only if it is a direct sum of cyclic subgroups of prime order. Proof. An aD-group is an aNC-group. For the converse, we show that an aNCgroup K has trivial commutator subgroup and so is abelian. If E Ÿ K is abelian, then K œ R z E for some R ü K and so Kw œ ÒR Eß R EÓ Ÿ R Hence, Kw is in the normal complement of any abelian subgroup of K , including all cyclic subgroups Ø+Ù of K. Hence, Kw is trivial.…

aC-Groups Finally, we state without proof the following theorem on aC-groups. Theorem 5.18 (aC-groups) 1) (Hall, P. [17], 1937; see also Schmidt [30], p. 122) A finite group K is an aC-group if and only if K is a direct sum of groups of square-free order. 2) (See Schmidt [30], p. 123) A group K is an aC-group if and only if K œ Š  N3 ‹ z Œ  L j 3−M

j−J

where each R3 and each L4 is cyclic of prime order and R3 ü K for all 3.…

Behavior Under Direct Sum The following theorem describes how some basic constructions behave with respect to direct sums and emphasizes the fact that the summands in a direct sum have a great deal of independence.

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Theorem 5.19 Let K œ  L3 3−M

Then the following hold: 1) (Center of K) ^ÐKÑ œ  ^ÐL3 Ñ 3−M

2) (Commutator of K) Kw œ  L3w 3−M

3) If R3 ü L3 for all 3, then  L3

3−M

 R3

3−M

¸ {

3−M

L3 R3

4) If L3 « K, then AutÐKÑ ¸ } AutÐL3 Ñ 3−M

Proof. For part 1), since the L3 's commute elementwise, it is clear that  ^ÐL3 Ñ Ÿ ^ÐKÑ. But if D − ^ÐKÑ, then let D œ 2" â28 with each 23 in a different factor L43 . Then D − ^ÐL3 Ñ and 25 − ^ÐL3 Ñ for 5 Á 3 and so 23 − ^ÐL3 Ñ, whence D −  ^ÐL3 Ñ. For part 2), Theorem 3.41 implies that Kw œ Ò  L3 ß  L4 Ó œ  ÒL3 ß L4 Ó œ  ÒL3 ß L3 Ó œ  L3w 3

4

3ß4

3

3

For part 3), the function 5À  L3 Ä { 3−M

3−M

L3 R3

defined by 5Ð+ÑÐ5Ñ œ ,5 Ð+ÑR5 œ 1R5 ‰ ,5 Ð+Ñ where 1R5 is the canonical projection map and ,5 is the 5 th injection map is an epimorphism. Moreover, 5+ œ " if and only if ,5 Ð+Ñ − R5 for all 5 and so kerÐ5Ñ œ  R3 . As to part 4), since L3 « K, if 5 − AutÐKÑ, then 5lL3 − AutÐL3 Ñ and so we can define a map 0 À AutÐKÑ Ä } 3−M AutÐL3 Ñ by

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0 Ð5ÑÐ3Ñ œ 5lL3 This is a group homomorphism since 0 Ð57 ÑÐ3Ñ œ Ð5 ‰ 7 ÑlL3 œ Ð5lL3 Ñ ‰ Ð7 lL3 Ñ œ 0 Ð5ÑÐ3Ñ ‰ 0 Ð7 ÑÐ3Ñ œ Ò0 Ð5Ñ0 Ð7 ÑÓÐ3Ñ for all 3 − M and so 0 Ð5 ‰ 7 Ñ œ 0 Ð5Ñ0 Ð7 Ñ. Furthermore, 0 is injective since 5lL3 œ + for all 3 implies 5 œ + . To see that 0 is surjective, if 7 − } 3−M AutÐL3 Ñ, then the universality of the direct sum implies that there is a unique homomorphism 5À K Ä K satisfying 5lL3 œ 73 , the 3th coordinate of 7 . But the bijectivity of each 73 implies that 5 − AutÐKÑ and so 0 Ð5Ñ œ 7 .…

When All Subgroups Are Normal All subgroups of an abelian group are normal, but all subgroups of the quaternion group U are normal and yet U is not abelian. A Hamiltonian group is a nonabelian group all of whose subgroups are normal. In 1933, Baer [3] published a characterization of Hamiltonian groups. Baer's theorem says that the Hamiltonian groups are actually a special type of abelian group with an additional quaternion direct summand. Theorem 5.20 (Baer [3], 1933) A group K is Hamiltonian if and only if KœUEF where U is a quaternion group, E is an elementary abelian group of exponent # and F is an abelian group all of whose elements have odd order. Proof. First suppose that K œ U  E  F and let L Ÿ K. If L Ÿ E  F , then L ü K. Note that if " − L , then Kw œ Ö„"× Ÿ L and so L ü K . Every 2 − L has the form 2 œ ;+, where ; − U, + − E and , − F has odd order, then 2#9Ð,Ñ œ ; #9Ð,Ñ œ ; # and so ; # − L . If 9Ð;Ñ œ % for some 2 − L , then " − L and so L ü K . On the other hand, if 9Ð;Ñ Ÿ # for all 2 − L , then L Ÿ Ö„"×  E  F œ ^ÐKÑ and so L ü K. Thus, K is Hamiltonian. For the converse, let K be Hamiltonian. Theorem 3.38 shows that K is periodic and that any nonabelian subgroup of K contains a quaternion subgroup. If F is the set of odd-order elements of K and Q is the set of elements of order a power of #, then F ∩ Q œ Ö"×. Moreover, for any Bß C − K, the normality of ØBÙ and ØCÙ imply that ØBß CÙ œ ØBÙØCÙ and so 9ÐBCÑ ± 9ÐBÑ9ÐCÑ . It follows that

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F and Q are (normal) subgroups of K. Finally, every + − K has order 9Ð+Ñ œ #5 7 where 7 is odd and so Corollary 2.11 implies that + œ ,7 for some , − F and 7 − Q . Thus, KœFQ Since F does not contain a quaternion subgroup, it must be abelian and since Q is therefore nonabelian (since K is nonabelian), Q contains a quaternion subgroup U œ ØBß CÙ. We are left with showing that Q œ U  E, where E is an elementary abelian group of exponent #. The centralizer G œ GQ ÐUÑ of U œ ØBß CÙ in Q has exponent #. To see this, note that if D − G has order %, then ØDBÙ ü Q has order % and so DB$ œ ÐDBÑC œ DB

or DB$ œ ÐDBÑC œ D $ B$

and since the former is false, we have D # œ ". Thus, G is an elementary abelian group of exponent #. Moreover, since G is a vector space over ™# , every subgroup (i.e., subspace) has a direct complement and so G œ ØB# Ù  E where E is also an elementary abelian group of exponent #. Consider the quotient Q ÎG , which contains the four distinct cosets G , BG , CG and BCG . Poincaré's theorem gives ÐQ À GÑ œ ÐQ À GQ ÐBÑ ∩ GQ ÐCÑÑ Ÿ ÐQ À GQ ÐBÑÑÐQ À GQ ÐCÑÑ œ ¸BQ ¸¸CQ ¸ œ% and so Q ÎG œ ÖGß BGß CGß BCG× which implies that Q œ UG œ UÐØB# Ù  EÑ œ UE But the only involution in U is B# , which is not in E and so U ∩ E œ Ö"×. Thus, Q œ U  E and KœQ FœUEF

…

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Semidirect Products We now turn to semidirect products. Definition Let K be a group. If K œ R ì L, R ü K then R is called a normal complement of L in K and K is called the semidirect product of R by L , denoted by K œR zL A semidirect product is nontrivial if both factors are proper.… Example 5.21 The dihedral group H#8 is a nontrivial semidirect product: H#8 œ Ø3Ù z Ø5Ù The symmetric group is also a nontrivial semidirect product: W8 œ E8 z ØÐ" #ÑÙ On the other hand, the quaternion group is not a nontrivial semidirect product, since the orders of the factors must be # and % but the only subgroup of U of order # is contained in every subgroup of order %.… For an arbitrary semidirect product K œ R z L , the commutativity rule is 28 œ Ð282" Ñ2 œ 82 2 for 2 − L and 8 − R and this yields the multiplication rule Ð8" 2" ÑÐ8# 2# Ñ œ 8" 82# " 2" 2# for 23 − L and 83 − R . Thus, for the semidirect product, the multiplication rule shows that “cross products” are involved, in the form of conjugates 82" " . This has some perhaps unexpected consequences. For example, if E" ¸ E# and F" ¸ F# , then E"  F" ¸ E#  F# . On the other hand, if Ø+Ù is cyclic of order ', then Ø+Ù œ Ø+# Ù z Ø+$ Ù œ Ö"ß +# ß +% ×  Ö"ß +$ × But we also have W$ œ ØÐ" # $ÙÙ z ØÐ" #ÑÙ œ Ö+ß Ð" # $Ñß Ð" $ #Ñ× z Ö+ß Ð" #Ñ× and so the nonisomorphic groups Ø+Ù and W$ can be written as a semidirect product, where corresponding factors are isomorphic! The reason that this is not a contradiction is that the values of the conjugates are different in each group. For instance,

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$

Ð+# Ñ+ œ +# but Ð" # $ÑÐ" #Ñ œ Ð# " $Ñ Á Ð" # $Ñ

Semidirect Products and Extensions of Epimorphisms Let K and K" be groups with L Ÿ K and let 5À L q» K" be an epimorphism. The key to describing the possible extensions of 5 to K is to consider the possible kernels for such an extension. Suppose that K is a group and that Lß O Ÿ K contain a normal subgroup N ü K. Then LÎN and OÎN are complements in KÎN if and only if L ∩O œN

and K œ LO

It will be convenient to make the following nonstandard definition. Definition Let K be a group and let N ü K. If Lß O Ÿ K satisfy L ∩O œN

and K œ LO

we will say that L and O are complements modulo N . If O is a normal complement of L modulo N , we will write K œ O z L Òmod N Ó

…

Now, if 5À K q» K" is an extension of 5À L q» K" , then L ∩ kerÐ5Ñ œ kerÐ5 Ñ To see that K œ LkerÐ5Ñ, since 5 is surjective, for any + − K, there is an 2 − L for which 5+ œ 52 œ 52 and so 2" + − kerÐ5Ñ, whence + œ 2Ð2" +Ñ − L kerÐ5Ñ and so kerÐ5Ñ is a normal complement of L modulo kerÐ5Ñ: K œ kerÐ5Ñ z L Òmod kerÐ5ÑÓ

(5.22)

It follows that every extension 5 of 5 is uniquely determined by its kernel kerÐ5Ñ. On the other hand, suppose that K œ O z L Òmod kerÐ5ÑÓ Then the ignore-O map 5À K Ä K" defined by 5Ð25Ñ œ 5Ð2Ñ

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for 2 − L and 5 − O is well defined since if 25 œ 2" 5" for 2" − L and 5" − O , then 2"" 2 œ 5" 5 " − L ∩ O œ kerÐ5Ñ and so 5Ð2"" 2Ñ œ ", that is, 52 œ 52" . The normality of O implies that 5 is a group homomorphism, since 5Ð525" 2" Ñ œ 5Ð55"2 22" Ñ œ 5Ð22" Ñ œ 5Ð2Ñ5Ð2" Ñ œ 5Ð52Ñ5Ð5"2"Ñ The kernel of 5 is kerÐ5Ñ œ Ö25 ± 52 œ !× œ kerÐ5ÑO œ O and so 5 is the unique extension of 5 with kernel O . Theorem 5.23 Let K and K" be groups, let L Ÿ K and let 5À L q» K" be an epimorphism. 1) Given any normal complement O of L modulo kerÐ5Ñ, K œ O z L Òmod kerÐ5ÑÓ the ignore-O map 5O À K Ä K" defined by 5Ð25Ñ œ 52 for all 2 − L and 5 − O is the unique extension of 5 with kernel O . Moreover, every extension 5 of 5 is an ignore-O map, where O œ kerÐ5 Ñ. 2) If K œ O z L , then 5 has a unique extension 5 À K Ä K" for which kerÐ5Ñ œ O kerÐ5Ñ Proof. For part 2), the Dedekind law implies that L ∩ O kerÐ5Ñ œ kerÐ5ÑÐL ∩ OÑ œ kerÐ5Ñ and so O kerÐ5Ñ is a normal complement of L modulo kerÐ5Ñ.…

Semidirect Products and One-Sided Invertibility Semidirect products are related to one-sided invertibility. Definition Let 5À K Ä L be a group homomorphism. 1) A left inverse of 5 is a homomorphism 5P À L Ä K for which 5P ‰ 5 œ + . If 5 has a left inverse, then 5 is said to be left invertible. 2) A right inverse of 5 is a homomorphism 5V À K Ä L for which 5 ‰ 5V œ + . If 5 has a right inverse, then 5 is said to be right invertible.… Unlike the two-sided inverse, one-sided inverses need not be unique. A leftinvertible homomorphism 5 is injective, since

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5+ œ 5,

Ê

5P ‰ 5 + œ 5P ‰ 5 ,

Ê

+œ,

and a right-invertible homomorphism 5 is surjective, since if , − L , then , œ 5Ð5V ,Ñ − imÐ5Ñ For set functions, the converses of these statements hold: 5 is left-invertible if and only if it is injective and 5 is right-invertible if and only if it is surjective. However, this is not the case for group homomorphisms. Let 5À K ä K" be injective. Referring to Figure 5.3,

K σ|

im(σ)

im(σ) (σ|

)

im(σ) -1

G

G1

Figure 5.3 the map 5limÐ5Ñ À K ¸ imÐ5Ñ obtained from 5 by restricting its range to imÐ5Ñ is an isomorphism and the left inverses of 5 are precisely the extensions of 5P œ Ð5limÐ5Ñ Ñ" À imÐ5Ñ ¸ K to K" . Hence, Theorem 5.23 implies that there is one left inverse 5P for 5 for each normal complement O of imÐ5Ñ and kerÐ5P Ñ œ O . Moreover, this accounts for all left inverses of 5 . In particular, 5 is left-invertible if and only if imÐ5Ñ has a normal complement. Now let 5À K q» K" be surjective. Referring to Figure 5.4,

ker(σ) H

σ|H

G

σR=(σ|H)-1

G1

Figure 5.4 If K œ kerÐ5Ñ z L for some L Ÿ K, then 5lL À L ¸ K" and Ð5lL Ñ" is a right inverse of 5 , with image L . Conversely, if 5V À K" ä K is a right inverse of 5, then K œ kerÐ5Ñ z imÐ5V Ñ For if + − kerÐ5Ñ ∩ imÐ5V Ñ, then + œ 5V , and 5+ œ ", whence , œ " and so

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+ œ ". Also, for any + − K, we have

+ œ ˆ+ÒÐ5V ‰ 5Ñ+Ó" ‰ Ð5V ‰ 5Ñ+ − kerÐ5Ñ z imÐ5V Ñ

Theorem 5.24 Let 5À K Ä K" be a group homomorphism. 1) If 5 is injective, then 5 has a unique left inverse 5P with kernel O for each normal complement O of imÐ5Ñ. Hence, K" œ kerÐ5P Ñ z imÐ5Ñ This accounts for all left inverses of 5 . 2) If 5 is surjective, then 5 has a unique right inverse 5V œ Ð5lL Ñ" À K" Ä K for each complement L of kerÐ5Ñ. Hence, L œ imÐ5V Ñ and K œ kerÐ5Ñ z imÐ5V Ñ This accounts for all of the right inverses of 5.… Here is a nice application of this theorem. Theorem 5.25 Let L and O be indecomposable groups. Let αÀ O Ä L and " À L Ä O be homomorphisms for which α" − AutÐLÑ and imÐ" Ñ ü O . Then α and " are isomorphisms. Proof. The map "P œ Ðα" Ñ" ‰ αÀ O Ä L is a left inverse of " and so " is injective and L œ kerÐ"P Ñ  imÐ" Ñ But since "P is not the zero map, " must be surjective as well. Hence, " and therefore also α are isomorphisms.…

The External Semidirect Product To see how to externalize the semidirect product, let us review the internal version. If we write the multiplication rule for the semidirect product R z L in the form Ð8" 2" ÑÐ8# 2# Ñ œ 8" Ò#2" Ð8# ÑÓ2" 2# where #2 is conjugation by 2, then R z L is completely described by the subgroups R and L , along with the family \ œ Ö#2 ± 2 − L× of inner automorphisms. Moreover, since the map # À 2 È #2 is a homomorphism, # is a representation of L in AutÐR Ñ. In the spirit of externalization, we separate the components by writing 25 as Ð2ß 5Ñ to get

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Ð8" ß 2" ÑÐ8# ß 2# Ñ œ Ð8" #2" Ð8# Ñß 2" 2# Ñ But if the factors R and L are to be arbitrary groups, then the inner automorphisms #2 make no sense. However, any representation of )À L Ä AutÐR Ñ of L in AutÐR Ñ can play the role of conjugation and define a group, as we now show. Theorem 5.26 Let R and L be groups and let ) À L Ä AutÐR Ñ be a homomorphism. We denote ) Ð2Ñ by )2 . Let R z ) L be the cartesian product R ‚ L together with the binary operation defined by Ð+ß BÑÐ,ß CÑ œ Ð+)B Ð,Ñß BCÑ Then K œ R z ) L is a group; in fact, it is a semidirect product R z ) L œ R ‚ Ö"× z Ö"× ‚ L Also, Ð+ß BÑ œ Ð+ß "ÑÐ"ß BÑ and Ð+ß "ÑÐ"ßBÑ œ Ð)B Ð+Ñß "Ñ which shows that ) does define the inner automorphisms of elements of R ‚ Ö"× by elements of Ö"× ‚ L . The group R z ) L is called the external semidirect product of R by L defined by ) . Proof. To see that multiplication is associative, we have ÒÐ+ß BÑÐ,ß CÑÓÐ-ß DÑ œ Ð+)B Ð,Ñß BCÑÐ-ß DÑ œ Ð+)B Ð,Ñ)BC Ð-Ñß BCDÑ and Ð+ß BÑÒÐ,ß CÑÐ-ß DÑÓ œ Ð+ß BÑÐ, )C Ð-Ñß CDÑ œ Ð+)B Ð, )C Ð-ÑÑß BCDÑ œ Ð+)B Ð,Ñ)BC Ð-ÑÑß BCDÑ It is easy to check that Ð"ß "Ñ is the identity and Ð+ß BÑ" œ Ð)B" Ð+" Ñß B" Ñ Thus, R z ) L is a group. For the rest, routine calculation gives Ð+ß "ÑÐ,ß "Ñ œ Ð+,ß "Ñ

and Ð+ß "Ñ" œ Ð+" ß "Ñ

Ð"ß BÑÐ"ß CÑ œ Ð"ß BCÑ

and Ð"ß BÑ" œ Ð"ß B" Ñ

and

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177

and so R ‚ Ö"× and Ö"× ‚ L are subgroups of R z ) L . To see that R ‚ Ö"× is normal, since Ð+ß BÑ œ Ð+ß "ÑÐ"ß BÑ we need only check that Ð+ß "ÑÐ"ßBÑ œ Ð"ß BÑÐ+ß "ÑÐ"ß B" Ñ œ Ð"ß BÑÐ+ß B" Ñ œ Ð)B Ð+Ñß "Ñ − R ‚ Ö"× and Ð+ß "ÑÐ,ß"Ñ œ Ð,ß "ÑÐ+ß "ÑÐ, " ß "Ñ œ Ð,+, " ß "Ñ − R ‚ Ö"× Clearly, ÐR ‚ Ö"×Ñ ∩ ÐÖ"× ‚ LÑ œ ÖÐ"ß "Ñ× and so R z ) L is the (internal) semidirect product of R ‚ Ö"× by Ö"× ‚ L .… Note that the zero representation )À L Ä AutÐKÑ defined by )Ð2Ñ œ +K defines the external direct product K } L . It is common to write the external version of the semidirect product using the notation of the internal version. Thus, if K and L are groups, we can specify a semidirect product W œ K z ) L by taking the set of formal products W œ Ö12 ± 1 − Kß 2 − L× in place of ordered pairs and specifying the commutativity rule 21 œ )2 Ð1Ñ2 where )À L Ä AutÐKÑ is a homomorphism. Example 5.27 Given a group K, there is one rather obvious way to create an external semidirect product K z ) L , namely, by taking L œ AutÐKÑ and )À L Ä L to be the identity. The group product in this case is Ð+5ÑÐ, 7 Ñ œ +5Ð,Ñ57 The group K z ) AutÐKÑ is called the holomorph of K. We may generalize this by taking L to be any subgroup of AutÐKÑ and ) to be the inclusion map from L to AutÐKÑ. The group K z ) L is called the relative holomorph of K with respect to L . Finally, if 5 − AutÐKÑ, then the relative holomorph K z ) Ø5 Ù is called the extension of K by 5. The group product in this case is Ð+53 ÑÐ, 54 Ñ œ +53 Ð,Ñ534 As an example, let K œ G#8 Ð+Ñ be cyclic of order #8 , where 8   $. If 7 œ #8" , then the power map defined by .+ œ +7" is an automorphism of K

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of order #, since 7  " is relatively prime to #8 and Ð7  "Ñ# œ Ð#8#  "Ñ#8  " ´ " mod #8 Thus, the extension of K by . is just WH8 œ G#8 Ð+Ñ z ) G# ÐBÑ and has group product Ð+3 ß "ÑÐ+4 ß "Ñ œ Ð+34 ß "Ñ Ð+3 ß BÑÐ+4 ß "Ñ œ Ð+3Ð7"Ñ4 ß BÑ Ð+3 ß "ÑÐ+4 ß BÑ œ Ð+34 ß BÑ Ð+3 ß BÑÐ+4 ß BÑ œ Ð+3Ð7"Ñ4 ß "Ñ for all ! Ÿ 3ß 4 Ÿ #8  ". Since Ð+3 ß B4 Ñ œ Ð+ß "Ñ3 Ð"ß BÑ4 , setting α œ Ð+ß "Ñ and 0 œ Ð"ß BÑ gives WH8 œ Øαß 0 Ùß

9ÐαÑ œ #8 ß

9Ð0Ñ œ #ß

0α œ α#

8"

"

0

The group WH8 is called the semidihedral group of order #8" .… Example 5.28 Recall that if G is an infinite cyclic group, then AutÐG Ñ œ Ö+ ß 7 × where 7 À + È +" and if G is cyclic of order 8, then AutÐGÑ ¸ ™8‡ . Let G∞ Ð+Ñ and G∞ Ð,Ñ be infinite cyclic groups. Then a homomorphism )À G∞ Ð,Ñ Ä AutÐG∞ Ð+ÑÑ œ Ö+ß 7 × is completely determined by the value ), , which can be either + or 7 . If ), œ + , then ) is the zero map and G∞ Ð+Ñ z ) G∞ Ð,Ñ is direct. If ), œ 7 , then the commutativity rule in the group G∞ Ð+Ñ z ) G∞ Ð,Ñ is ,+ œ 7 Ð+Ñ, œ +" , The automorphisms of a finite cyclic group G8 Ð+Ñ are the 5 th power homomorphisms 55 defined by 55 Ð+Ñ œ +5 where 5 − ™‡8 . Thus, the representations )5 À G∞ Ð,Ñ Ä AutÐG8 Ð+ÑÑ are the homomorphisms defined by )5 Ð,Ñ œ 55 and so the possible semidirect products are G8 Ð+Ñ z )5 G∞ Ð,Ñ œ Ö+3 , 4 ± ! Ÿ 3 Ÿ 8  "ß 4 − ™ß ,+ œ +5 ,× where 5 − ™‡8 .… Example 5.29 To define a semidirect product G$ Ð+Ñ z ) G% Ð,Ñ, we must specify a homomorphism )À G% Ð,Ñ Ä AutÐG$ Ñ œ Ö+ß 5# × where 5# Ð+Ñ œ +# . The zero homomorphism defines the direct product

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G$ Ð+Ñ } G% Ð,Ñ. If ), œ 5# , then the semidirect product X œ G$ Ð+Ñ z ) G% Ð,Ñ defined by ) is X œ Ø+ß ,Ùß

9Ð+Ñ œ $ß

9Ð,Ñ œ %ß

,+ œ +# ,

y E% since We have not yet encountered this group of order "#. In particular, X ¸ y H"# since 9Ð,Ñ œ %. We will show later that E% , H"# and X 9Ð+, # Ñ œ ' and X ¸ are the only nonabelian groups of order "# (up to isomorphism).… Example 5.30 Let us examine the possibilities for an external semidirect product of the form G:7 Ð+Ñ z ) G:8 Ð,Ñ where : is prime. The automorphisms of G:7 Ð+Ñ are the 5 th power maps ‡ 55 À + È +5 for 5 − ™7 . The function )À G:8 Ð,Ñ Ä AutÐG:7 Ð+ÑÑ satisfying ), œ 55 defines a homomorphism if and only if 9Ð55 Ñ ± 9Ð,Ñ, that is, if and only 8 if 55: œ +, or +5

Ð:8 Ñ

œ+

or finally 8

5 Ð: Ñ ´ " mod :7

(5.31)

As an example, for 7 œ ", Fermat's theorem implies that (5.31) is equivalent to 5 ´ " mod : and since " Ÿ 5  :, it follows that 5 œ ". Hence, the only semidirect product of the form G: Ð+Ñ z ) G:8 Ð,Ñ is the direct product. If 8 œ ", then (5.31) is equivalent to 5 : ´ " mod :7 Any 5 of the form 5 œ "  ?:7" where ?  : satisfies this congruence, since 5 : œ Ð"  ?:7" Ñ: œ "  A:7 Thus, for each ?  :, there is a semidirect product G:7 Ð+Ñ z ) G: Ð,Ñ where ,+ œ +"?:

7"

,

…

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Example 5.32 Let H be a group and let K œ H$ ³ H } H } H. Then each permutation 5 − W$ defines an automorphism )5 of K by permuting the coordinates in K. For example, )Ð" $Ñ ÐBß Cß DÑ œ ÐDß Cß BÑ Moreover, the map )À W$ Ä AutÐKÑ sending 5 to )5 is a homomorphism, since )5 )7 œ )57 . Thus, the semidirect product K z ) W$ œ H $ z ) W $ exists. To illustrate the product, if 5 œ Ð" $Ñ, then Ð+ß ,ß -Ñß 5Ñ ÐBß Cß DÑß 7 Ñ œ Ð+ß ,ß -Ñ)5 ÐBß Cß DÑß 57 œ Ð+ß ,ß -ÑÐDß Cß BÑß 57 œ Ð+Dß ,Cß -BÑß 57

…

We will generalize this example in the next section.

*The Wreath Product To generalize Example 5.32, let H be a group, let H be a nonempty set and let Kœ }H =−H

be the external direct product of kHk copies of H, indexed by H. Each permutation 5 of H defines an automorphism )5 of K by permuting the coordinate positions of any 0 − K. Specifically, the =th coordinate of 0 becomes the Ð5=Ñ-th coordinate of )5 Ð0 Ñ, that is, )5 Ð0 ÑÐ5=Ñ œ 0 Ð=Ñ, or equivalently, Ð)5 0 ÑÐ=Ñ œ 0 Ð5 " =Ñ Thus, )5 Ð0 Ñ œ 0 ‰ 5" The map )5 À K Ä K is easily seen to be an automorphism of K, since it is clearly bijective and )5 Ð0 1Ñ œ Ð0 1Ñ ‰ 5" œ Ð0 ‰ 5" ÑÐ1 ‰ 5" Ñ œ )5 Ð0 Ñ)5 Ð1Ñ Moreover, if U Ÿ WH , then the map )À U Ä AutÐKÑ defined by )5 œ )5 is a homomorphism, since )57 0 œ 0 ‰ Ð57 Ñ" œ 0 ‰ Ð7 " ‰ 5" Ñ œ )7 Ð0 Ñ ‰ 5" œ )5 )7 Ð0 Ñ Hence, the semidirect product K z ) U exists. It is easy to describe the commutativity rule in words: To place the factors in the product 50 in the reverse order, simply permute the coordinate positions of 0 using 5.

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Note that it is not essential that the second coordinates in the ordered pairs Ð0 ß 5Ñ − K z ) U be actual permutations of H as long as they act like permutations, that is, as long as there is a homomorphism -À U Ä WH . As is customary, we denote the permutation -; of H by ; itself. Thus, if U acts on H, then for each ; − U, the map ); À K Ä K defined by ); 0 œ 0 ‰ ; " is an automorphism of K and the map ) À U Ä AutÐKÑ defined by ); œ ); is a homomorphism. The semidirect product K z ) U of K by U defined by ) is one version of the wreath product of K by U. The other version comes by replacing the external direct product K œ } H by the external direct sum K œ { H. Definition Let H be a group, let H be a nonempty set and let U be a group acting on H. We denote the action of ; − U on = − H by ;=. 1) Let Kœ }H =−H

Define a homomorphism ) À U Ä AutÐKÑ by ); œ ); where ); 0 œ 0 ‰ ; " Then the semidirect product K z ) U is called the complete wreath product of H by U with index set H and base K. 2) If we replace K by the external direct sum Kœ {H =−H

the resulting semidirect product K z ) U is called the restricted wreath product of K by U with index set H and base K. A common notation for the wreath product is H › U. To emphasize the index set, we will write H › H U. There does not seem to be a standard notion to distinguish the two wreath products, so we use H › U for the restricted wreath product and H ›› U for the complete wreath product.… Note that if H and U are finite groups and H is a finite set, then

kH ›› H Uk œ ¸HH z U¸ œ ¸HH ¸kUk œ kHkkHk kUk

Example 5.33 (Regular wreath product) Let H and U be groups and let Kœ }H ;−U

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be the direct product of H indexed by the group U. Let U act on itself by left translation, that is, ); < œ ;<, that is, the action of U is the left regular representation. In this case, the complete wreath product H ›› U is called a regular wreath product, which we denote by H ›› < U. Thus, the product has the form Ð0 ß ;ÑÐ1ß <Ñ œ Ð0 Ð1 ‰ ; " Ñß ;<Ñ

…

Wreath Products as Permutations Under certain reasonable conditions, the elements of a wreath product can be thought of as permutations. Specifically, let [ œ H ›› H U Let K œ } =−H H and assume that U acts faithfully on H. Now suppose that the group H acts faithfully on a nonempty set A. Then the elements of [ act on the set A ‚ H. In particular, for Ð0 ß ;Ñ − [ define Ð0 ß ;ч À A ‚ H Ä A ‚ H by Ð0 ß ;ч Ð-ß =Ñ œ Ð0 Ð; =Ñ-ß ; =Ñ The map Ð0 ß ;ч is injective since Ð0 ß ;ч Ð-ß =Ñ œ Ð0 ß ;ч Ð-w ß =w Ñ implies that Ð0 Ð; =Ñ-ß ; =Ñ œ Ð0 Ð; =w Ñ-w ß ; =w Ñ and so =w œ = and -w œ -. Also, Ð0 ß ;ч is surjective since for any Ð-ß =Ñ − A ‚ H, we have Ð0 ß ;ч Ð0 Ð=Ñ" -ß ; " =Ñ œ Ð-ß =Ñ Hence, Ð0 ß ;ч is a permutation of A ‚ H. Moreover, the map 5À [ Ä WA‚H defined by 5Ð0 ß ;Ñ œ Ð0 ß ;ч is a homomorphism, since ÒÐ0 ß ;ÑÐ1ß <ÑÓ‡ Ð-ß =Ñ œ Ð0 Ð1 ‰ ; "Ñß ;<чÐ-ß =Ñ œ ÐÒ0 Ð1 ‰ ; " ÑÐ;<=ÑÓ-ß ;<=Ñ œ ÐÒ0 Ð;<=ÑÐ1 ‰ ; " ÑÐ;<=ÑÓ-ß ;<=Ñ œ Ð0 Ð;<=Ñ1Ð<=Ñ-ß ;<=Ñ œ Ð0 ß ;ч Ð1Ð<=Ñ-ß <=Ñ œ Ð0 ß ;ч Ð1ß <ч Ð-ß =Ñ As to the kernel of 5, if Ð0 ß ;ч œ +, then

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Ð0 Ð; =Ñ-ß ; =Ñ œ Ð-ß =Ñ for all - − H and = − H. Since the actions of U on H and H on A are faithful, we deduce that ; œ " and 0 Ð=Ñ œ " for all =. Thus, 5À [ ä WA‚H is an embedding of [ into WA‚H . When [ œ H › H U is the restricted wreath product, we can describe the image of the embedding explicitly. For each . − H and α − H, let .α − K be defined by .α Ð=Ñ œ œ

. "

if = œ α if = Á α

Let \ be the subgroup of WA‚H generated by the permutations Ð.α ß "ч and Ð"ß ;ч , that is, \ œ ØÐ.α ß "ч ß Ð"ß ;ч ± . − Hß α − Hß ; − UÙ Certainly, \ © imÐ5Ñ. To see that the reverse inclusion holds, we observe that Ð0 ß ;ч œ ÒÐ0 ß "ÑÐ"ß ;ÑÓ‡ œ Ð0 ß "ч Ð"ß ;ч and so it is sufficient to show that Ð0 ß "Ñ − \ for all 0 − K . Since Ð0 1ß "ч œ ÒÐ0 ß "ÑÐ1ß "ÑÓ‡ œ Ð0 ß "ч Ð1ß "ч we have Ð0" â08 ß ;ч œ Ð0" ß "ч âÐ08 ß "ч Ð"ß ;ч Now, any 0 − K has finite support suppÐ0 Ñ œ Ö=" ß á ß =8 × and so 0 œ 0 Ð=" Ñ=" â0 Ð=8 Ñ=8 Hence, Ð0 ß "ч œ Ð0 Ð=" Ñ=" â0 Ð=8 Ñ=8 ß "ч œ Ð0 Ð=" Ñ=" ß "ч âÐ0 Ð=8Ñ=8 ß "ч − \ Thus, imÐ5Ñ œ \ . Theorem 5.34 Let [ œ H ›› H U be a wreath product and suppose that U acts faithfully on H. Suppose also that H acts faithfully on A. Then the map 5À [ Ä WA‚H defined by 5Ð0 ß ;Ñ œ Ð0 ß ;ч where Ð0 ß ;ч À A ‚ H Ä A ‚ H is defined by Ð0 ß ;ч Ð-ß =Ñ œ Ð0 Ð; =Ñ-ß ; =Ñ

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Fundamentals of Group Theory

is an embedding of [ into WA‚H . When [ œ H › H U is the restricted wreath product, the image of 5 is imÐ5Ñ œ ØÐ.α ß "ч ß Ð"ß ;ч ± . − Hß α − Hß ; − UÙ

…

It is convenient to drop the ‡ notation and to think of elements of a wreath product H ›› U as permutations of A ‚ H. Example 5.35 A permutation matrix T is an 8 ‚ 8 matrix with entries from the set Ö!ß "× with the property that each row contains exactly one " and each column contains exactly one ". Multiplication of a matrix E on the left by a permutation matrix T permutes the rows of E. Similarly, multiplication on the right by T permutes the columns of E. Let c8 be the multiplicative group of all 8 ‚ 8 permutation matrices. For T − c8 , let T3 denote the 3th row of T and let T Ð4Ñ denote the 4th column. The rows of T also define a permutation 1T of H œ Ö"ß á ß 8×, in particular, 1T Ð3Ñ is the column number of the " in row T3 . In this way, T8 is isomorphic to the symmetric group W8 . Clearly, the map 0 À T È 1T is bijective. If U − c8 , then ÐUT Ñ3ß4 œ U3 T Ð4Ñ œ " if and only if the column number 5 of the " in U3 is the same as the row number 5 of the " in column T Ð4Ñ , that is, if and only if 5 œ 1U Ð3Ñ implies that 1T Ð5Ñ œ 4. Hence, 1UT Ð3Ñ œ 4 implies that 1T Ð1U Ð3ÑÑ œ 4, that is, 1UT œ 1T 1U . Hence, 0 is an anti-isomorphism from c8 to W8 . Since the transpose map 7 À c8 Ä c8 is an anti-automorphism of c8 , it follows that the composite map 0 ‰ 7 is an isomorphism from c8 to W8 . We can generalize the notion of a permutation matrix as follows. If L is an abelian multiplicative group, define c8 ÐLÑ to be the set of 8 ‚ 8 matrices with the property that each row and each column has exactly one entry from L , all other entries being !. Matrix multiplication is defined using the product in L along with ! † ! œ ! † + œ ! and !  + œ +  ! œ + for + − L . This makes c8 ÐLÑ a group. Note that c8 is a subgroup of c8 ÐLÑ, with the identity of L playing the role of ". We can describe c8 ÐLÑ in terms of wreath products as follows. Let W8 act on H œ Ö"ß á ß 8× by the usual evaluation and let c8 act on H by T Ð5Ñ œ 1T Ð5Ñ. Let W be the set of all diagonal matrices in c8 ÐLÑ. Then every matrix in c8 ÐLÑ is a product HT where H − W and T − c8 . Also, W is a normal subgroup of c8 ÐLÑ. Hence, c8 ÐLÑ œ W z c8 . Now, any H − W can be identified with the ordered 8-tuple of diagonal elements of H and, in fact, W is isomorphic to L 8 , since matrix multiplication in W is elementwise product in L 8 . Hence, c8 ÐLÑ ¸ L 8 z c8 œ L › H c8 ¸ L › H W8

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185

When L is the group of 7th roots of unity in ‚, the group c8 ÐLÑ is called a generalized symmetric group or monomial group.…

Exercises 1.

Prove that the external direct product is commutative and associative, up to isomorphism, that is, L }O ¸O }L and ÐL } OÑ } P ¸ L } ÐO } PÑ

2.

Is there an identity for the external direct product? a) Suppose that K œ { K3 . For each 5 , let L5 œ Ö0 − { K3 ± 0 Ð3Ñ œ " for 3 Á 5×

3.

Show that L5 ü K and that K œ  L5 . b) If K œ  K3 , show that K ¸ { K3 . Let L , L3 and O be groups and let L œ L" } â } L= Suppose that 5À L ¸ O . For each 3, let L 3 œ Ö"× } â } Ö"× } L3 } Ö"× } â } Ö"× where L3 is in the 3th component. Prove that O œ 5L "  â  5L =

Let K œ L" âL8 be a finite group, where L3 ü K and Ð9ÐL3 Ñß 9ÐL4 ÑÑ œ " for 3 Á 4. Prove that the join 1L3 is a direct product, that is, K œ L"  â  L8 . 5. Suppose that K œ L  O and that R ü K. Prove that if R ∩ L œ Ö"× œ R ∩ O , then R Ÿ ^ÐKÑ. 6. Let 9ÐKÑ œ 78 where 7 and 8 are relatively prime. Let L Ÿ K with 9ÐLÑ œ 7. Show that a subgroup O Ÿ K is a complement of L if and only if 9ÐOÑ œ 8. 7. Prove that all nonabelian groups of order :$ , : prime are indecomposable. You may assume that all groups of order :# are abelian (which is true). 8. Prove that the group  of rational numbers is indecomposible. 9. Prove that H#8 is indecomposable if and only if 8 ´ y # mod %. 10. Let K œ Ø+Ù be a cyclic group. a) If K is infinite, show that K œ L ì O implies that L œ Ö"× or O œ Ö"×. b) If 9ÐKÑ œ 8, describe precisely the conditions under which a nontrivial essentially disjoint product representation K œ L ì O exists. 11. Let K œ  3−M L3 and let O Ÿ K . 4.

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Fundamentals of Group Theory a)

Show that it is not necessarily true that ?

O œ  ÐL3 ∩ OÑ 3−M

even if O ü K. b) Recall that a group that is equal to its own commutator subgroup is called perfect. Prove that if O is normal and perfect, then O œ  ÐL3 ∩ OÑ 3−M

c)

Prove that if K is periodic and if for all B −  L4 4Á3

and C − L3 , the orders 9ÐBÑ and 9ÐCÑ are relatively prime, then O œ  ÐL3 ∩ OÑ 3−M

This holds in particular for finite families if the factors L3 have relatively prime exponents. 12. Let K be a group and let L be a simple subgroup of K with index #. What can you say about any other nontrivial proper normal subgroup of K? Must such a subgroup exist? (For the latter, you may assume that the alternating group E& is simple and that W8 is centerless for 8   $.) 13. (Chinese remainder theorem for groups) Let K be a group and let L" ß á ß L8 be normal subgroups. Consider the map 5À K Ä } KÎL3 defined by 5+ œ Ð+L" ß á ß +L8 Ñ a) Show that 5 is a homomorphism with kerÐ5Ñ œ L" ∩ â ∩ L8 . b) Show that if the indices ÐK À L3 Ñ are finite and pairwise relatively prime, then 5 is surjective and so for any +" ß á ß +8 − K, there is a 1 − K for which 1 − ++3 L3 . This can also be written 1 ´ +" mod L" ã 1 ´ +8 mod L8 c) Discuss the uniqueness of the solution 1 in part b). 14. Let K be a group and let R be minimal normal in K . a) Prove that if K is characteristically simple, then there is a E © AutÐKÑ such that K œ R  Œ  5R 5−E

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187

Also, R is simple, as is every term in the sum above and so K is the direct sum of isomorphic simple subgroups. b) Prove that if K has the DCC on normal subgroups, then R is the direct sum of isomorphic simple groups. 15. To see that infinite groups are not, in general, cancellable in direct products, prove that ™ÒBÓ } ™ } ™ ¸ ™ÒBÓ } ™, where ™ÒBÓ is the abelian group of y ™. polynomials over ™ but ™ } ™ ¸ 16. Let K œ  W3 , where W3 is simple for all 3. Prove that the center ^ÐKÑ is the direct sum of those factors W3 that are abelian. Hence, K is centerless if and only if all of the factors W3 are nonabelian. 17. Let K œ  W3 3−M

be centerless, where each W3 is simple. Prove that the only minimal normal subgroups of K are the summands W3 . 18. Let K œ  W3 3−M

be centerless, where each W3 is simple. Prove that the normal subgroups of K are precisely the direct sums of the W3 's taken over the subsets of M . 19. Let K œ  3−M L3 where L3 are simple subgroups and L3 ¸ L4 for all 3ß 4 − M . Prove that K is characteristically simple. Hint: Consider the centers of the L3 .

Minimal Normal Subgroups 20. Let K be a group. Show that if L and O are distinct minimal normal subgroups of K, then L and O commute elementwse. 21. Let E be an abelian minimal normal subgroup of a group K. Show that if K œ EL where L  K, then E ∩ L œ Ö"×. 22. Let R" ß á ß R8 be minimal normal subgroups of K, let Q œ R" âR8 and let O ü K. Prove that there is a subset of R" ß á ß R8 , say after reindexing R" ß á ß R7 , such that OQ œ O  R"  â  R7 23. Let E Ÿ F Ÿ K and assume that E is a minimal normal subgroup of F and F is a minimal normal subgroup of K. Assume further that the set ÖE× œ Ö1E1" ± 1 − K× of conjugates of E is finite. Show that E is simple and that F is the direct product of conjugates of E. 24. Prove that the epimorphic image of a minimal normal subgroup is either trivial or minimal normal.

Semidirect Products 25. Let K œ L z O . Show that if 9ÐLÑ œ #, then K œ L  O . 26. Let K œ L z ) O . Show that if O is simple, then 9ÐOÑ ± 9ÐAutÐLÑÑ.

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27. Show that for every positive integer of the form 8 œ #Ð#5  "Ñ there is a centerless group of order 8. 28. Prove that if K œ L z O , then K œ L z O + for any + − K. Hence, if O is a complement of a normal subgroup, then so is any conjugate of O . 29. Show that it is not always possible to extend a homomorphism 5À L Ä Kw from a subgroup L Ÿ K to K. 30. Show that if K œ L z O is an internal semidirect product, then K is isomorphic to an external semidirect product K ¸ L z ) O for some )À O Ä AutÐLÑ. 31. Prove that ™' œ E z F

and W$ œ G z H

y W$ . where E ¸ ™$ ¸ G and F ¸ ™# ¸ H, but clearly ™' ¸ 32. Prove that W8 œ E8 z Ö+ß Ð" #Ñ×. 33. a) Prove that H) ¸ E z F" , where E ¸ ™% and F" ¸ ™# . b) Prove that H) ¸ G z F# , where G ¸ Z and F# ¸ ™# . What does this say about semidirect product decompositions? 34. Prove that H#8 ¸ E z F , where E ¸ ™8 and F ¸ ™# . 35. What is wrong, if anything, with the following argument? Let K œ L z O . Then the projection maps 3E À K Ä L and 3O À K Ä O defined by 3L Ð25Ñ œ 2 and 3O Ð25Ñ œ 5 have kernel O and L , respectively, whence both L and O are normal subgroups of K and so K œ L  O . 36. Recall that K œ KPÐ8ß J Ñ is the general linear group of all invertible 8 ‚ 8 matrices over the field J and W œ WPÐ8ß OÑ is the subgroup of matrices with determinant equal to ". a) Prove that WPÐ8ß OÑ ü KPÐ8ß J Ñ. b) Find a complement of WPÐ8ß J Ñ in KPÐ8ß J Ñ. Hint: How does one get a special matrix from a general one? 37. Let K be a group. For any + − K, denote left translation by + by j+ . Thus, j+ À K Ä K is defined by j+ ÐBÑ œ +B. Let _ œ Öj+ ± + − K×. Let T œ AutÐKÑ. Finally, let L œ Ø_ß TÙ be the subgroup of the symmetric group WK generated by _ and T. a) Prove that L œ _ z T. b) Prove that GL Ð_Ñ œ e, the subgroup of all right translations <+ À B È B<. 38. In this exercise, we describe all groups of order :; where :  ; are primes. We will assume a fact to be proved later in the book: If K is a finite group and if : is the smallest prime dividing 9ÐKÑ, then any subgroup of index : is normal in K. a) Describe the automorphism group AutÐG: Ñ, where G: œ Ø+Ù is cyclic of order a prime :. b) Show that any group of order :; is a semidirect product of a group of order ; by a group of order :. c) Describe the possible external direct products of G; by G: , where : and ; are distinct primes.

Direct and Semidirect Products

189

d) “Internalize” the external semidirect products in the previous part to show that up to isomorphism, the groups of order :; are the direct product G;  G: and for each 7 Á " satisfying 7: ´ " mod ; , a group described by K œ Øαß " Ùß 9ÐαÑ œ :ß 9Ð"Ñ œ ;ß α" œ " 7 α 39. Let E œ Ö7#8 ± 7ß 8 − ™× be the additive subgroup of  and let F œ ™B be an additive infinite cyclic group. Prove that the group K œ E z ) F , where )B Ð+Ñ œ #+, has the ACC on normal subgroups but that the normal subgroup E of K does not have the ACC on normal subgroups.

Wreath Products 40. One must be cautious in working with the action of 5 − W8 on the product H8 . Recall that 5 permutes the coordinates of an element of H8 . Suppose that Ð." ß á ß .8 Ñ − H8 . Then 5Ð." ß á ß .8 Ñ œ Ð.5" ß á ß .58 Ñ For 7 − W8 , compute Ð75 ÑÐ." ß á ß .8 Ñ. Are you sure? 41. In this exercise, we take a combinatorial look at the wreath product [ œ W# › W& , with the help of Figure 5.5.

a1 b5 a5 b4

5

b1

0 4 a4

a2

1 2

3

b2 a3

b3

Figure 5.5 Informally speaking, a graph is a set of vertices (or nodes) together with a set of edges connecting pairs of vertices. Figure 5.6 is an example of a graph Z . Two vertices are said to be adjacent if there is an edge between them. A bijection of the vertices of a graph that preserves adjacency is called an isomorphism. Show that the automorphisms of Z are isomorphic to the wreath product [ œ W# › W& . 42. Show that the wreath product is associative. Specifically, let K act on ?, L act on H and O act on A, all actions being faithful. Show that ÐK ›› H LÑ ›› A O ¸ K ›› H ÐL ›› A OÑ. 43. Show that the regular wreath product is not associative. Specifically, if K, L and O are nontrivial finite groups, explain why ÐK › < LÑ › < O cannot possibly be isomorphic to K › < ÐL › < OÑ.

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44. Show that the restricted wreath product G# › G# is isomorphic to the dihedral group H) . 45. Let \ be a nonempty set of size 85 and let c œ ÖF" ß á ß F8 × be a partition of \ into equal-sized blocks of size 5 . Call a permutation 5 − W\ nice if it also permutes the blocks, that is, for all 3, there is a 4 such that 5F3 œ F4 . Show that the set R of nice permutations is a group isomorphic to W5 › W8 . 46. Referring to Example 5.35, let L œ Ö„"× be the multiplicative group of square roots of unity. The group c8 ÐLÑ is called the hyperoctahedral group. Show that c8 ÐLÑ is isomorphic to the subgroup K Ÿ W#8 of all permutations of \ œ Ö8ß á ß "ß "ß á ß 8× with the property that 1Ð5Ñ œ 1Ð5Ñ. 47. Let [ œ H ›› H U be a wreath product where H act faithfully on A. Suppose that both actions are transitive, that is, for any =ß =w − H there is a ; − U for which ; = œ =w and similarly for the other action. Prove that the permutation representation of [ is also transitive, that is, for any pair Ð-ß =Ñß Ð-w ß =w Ñ − A ‚ H there is an Ð0 ß ;Ñ − [ for which Ð0 ß ;ч Ð-ß =Ñ œ Ð-w ß =w Ñ

Chapter 6

Permutation Groups

Permutations are fundamental to many branches of mathematics. In this chapter, we examine permutations from a group-theoretic perspective.

The Definition and Cycle Representation A permutation of a nonempty set \ is a bijective function on \ . The set of all permutations of \ is denoted by W\ . As is customary, when \ œ M8 ³ Ö"ß á ß 8× we write W\ as W8 . As we have seen, the set W\ of permutations of a nonempty set \ forms a group under composition of functions. For the record, we have: Definition Let \ be a nonempty set. The symmetric group W\ on the set \ is the group of all permutations of \ , under composition of functions.… There are various notations for permutations. The notation 5œŒ

+" +3"

+# +3#

â +8 â +38

is sometimes used to denote the permutation that sends +4 in the top row to +34 in the bottom row. This notation is a bit combersome and we prefer the cycle notation described in an earlier chapter. In particular, if +3 Á +4 − \ for 3 Á 4, then 5 œ Ð+" â+5 Ñ denotes the permutation 5 − W\ defined, for ! Ÿ 3 Ÿ 8, by 5+3 œ œ

+3" +"

for " Ÿ 3 Ÿ 5  " for 3 œ 5

and sending all other elements of \ to themselves. Such a permutation 5 is called a 5 -cycle in W\ . A #-cycle S. Roman, Fundamentals of Group Theory: An Advanced Approach, DOI 10.1007/978-0-8176-8301-6_6, © Springer Science+Business Media, LLC 2012

191

192

Fundamentals of Group Theory

5 œ Ð+ ,Ñ is called a transposition. Two cycles Ð+" â+5 Ñ and Ð," â,7 Ñ are disjoint if +3 Á ,4 for all 3ß 4. When \ is an infinite set containing the elements Ö+3 ± 3 − ™×, then we can define the infinite cycle 5 œ Ðá ß +" ß +! ß +" ß á Ñ where +3 Á +4 for 3 Á 4 as the permutation 5 sending +5 to +5" for all 5 − ™ and leaving all other elements of \ fixed. Theorem 6.1 For a permutation group W\ , the following hold. 1) Disjoint cycles in W\ commute. 2) Every permutation in W\ is a product of disjoint cycles, the product being unique except for the order of factors. A representation of 5 as a product of disjoint cycles (with or without "-cycles) is called a cycle representation or cycle decomposition of 5 . The cycle structure of a permutation 5 is the sequence of cycle lengths in a cycle decomposition of 5, or equivalently, the number of cycles of each length in a cycle decomposition of 5 . Proof. Part 1) we leave to the reader. For part 2), let 5 − W\ and define an equivalence relation on \ by B ´ C if C œ 55 B for some integer 5 . For B − \ , the equivalence class containing B is ÒBÓ œ Ö53 B ± 3 − ™× Note that the restriction 5lÒBÓ is a permutation of ÒBÓ. In fact, if the elements 53 B are distinct for all 3 − ™, then 5lÒBÓ is the infinite cycle 5l ÒBÓ œ Ðá ß 5 # Bß 5 " Bß Bß 5Bß 5 # Bß á Ñ On the other hand, if 53 B œ 54 B for 3  4, then 543 B œ B and if 7 is the smallest positive integer for which 57 B œ B, then 5lÒBÓ is the 7-cycle 5lÒBÓ œ ÐBß 5Bß 5# Bß á ß 57" BÑ The distinct equivalence classes form a partition of \ and 5 is a product of the disjoint cycles 5lF as F varies over these equivalence classes.… A cycle of length 5 has order 5 in W\ . Thus, a transposition is an involution, as is any product of disjoint transpositions. A power of a cycle need not be a cycle; for example Ð" # $ %Ñ# œ Ð" $ÑÐ# %Ñ Since disjoint cycles commute, if 5 œ -" â-7 is a cycle representation of 5, then for any integer 5 ,

Permutation Groups

193

5 55 œ -"5 â-7

Moreover, 55 œ + if and only if -35 œ + for each 3. Theorem 6.2 The order of 5 − W\ is the least common multiple of the lengths (orders) of the disjoint cycles in a cycle decomposition of 5.… The group properties of W\ do not depend upon the nature of the set \ , but only upon its cardinality. More formally put, if k\ k œ k] k, then the groups W\ and W] are isomorphic. Accordingly, we will feel free to state theorems in terms of either W\ or W8 (when \ is finite).

A Fundamental Formula Involving Conjugation When a permutation 7 − W\ is written as a product of disjoint cycles, it is very easy to describe the conjugates 7 5 of 7 . It is also remarkably easy to tell when two permutations are conjugate. Theorem 6.3 1) Let 5 − W8 . For any 5 -cycle Ð+" â+5 Ñ, Ð+" â+5 Ñ5 œ Ð5+" â5+5 Ñ Hence, if 7 œ -" â-5 is a cycle decomposition of 7 , then 7 5 œ -"5 â-55 is a cycle decomposition of 7 5 . 2) Two permutations are conjugate if and only if they have the same cycle structure. Proof. For part 1), we have Ð+" â+5 Ñ5 Ð5+3 Ñ œ œ

5+3" 5+"

35 3œ5

Also, if , Á 5+3 for any 3, then 5" , Á +3 and so Ð+" â+5 Ñ5 , œ 5Ð+" â+5 ÑÐ5" ,Ñ œ 5Ð5" ,Ñ œ , Hence, Ð+" â+5 Ñ5 is the cycle Ð5+" â5+5 Ñ. For part 2), if 7 œ -" â-7 is a cycle decomposition of 7 , then 5 7 5 œ -"5 â-7

and since -35 is a cycle of the same length as -3 , the cycle structure of 7 5 is the same as that of 7 . For the converse, suppose that 5 and 7 have the same cycle structure. If 5 and 7 are cycles, say

194

Fundamentals of Group Theory

5 œ Ð+" â +8 Ñ

and

7 œ Ð," â ,8 Ñ

then any permutation - that sends +3 to ,3 satisfies 5 - œ 7 . More generally, if 5 œ -" â-7

and

7 œ ." â.7

are the cycle decompositions of 5 and 7 , ordered so that -5 has the same length as .5 , then we can define a permutation - that sends the element in the 3th position of -5 to the element in the 3th position of .5 . Then 5- œ 7 .… The previous theorem implies that it is easy to tell when a subgroup L of the symmetric group W\ is normal. Theorem 6.4 A subgroup L Ÿ W\ is normal if and only if whenever 5 − L , then so are all permutations in W\ with the same cycle structure as 5 .…

Parity Every cycle is a product of transpositions, to wit Ð+" +# â+7 Ñ œ Ð+" +7 ÑÐ+" +7" ÑâÐ+" +$ ÑÐ+" +# Ñ and so every permutation is a product of transpositions. While this representation is far from unique, the parity of the number of transpositions is unique. Theorem 6.5 Let 5 − W8 . 1) Exactly one of the following holds: a) 5 can be written as a product of an even number of transpositions, in which case we say that 5 is even (or has even parity). b) 5 can be written as a product of an odd number of transpositions, in which case we say that 5 is odd (or has odd parity). 2) The symbol Ð"Ñ5 or sgÐ5Ñ, called the sign or signum of 5 , is equal to " if 5 is even and " if 5 is odd. We have Ð"Ñ5 œ Ð"Ñ85 where 5 is the number of cycles in the cycle decomposition of 5 (including "-cycles). Proof. For part 1), if 5 can be written as a product of an even number of transpositions and an odd number of transpositions, say 5 œ 3" â3#@ œ 7" â7#?" then the identity can be written as a product of an odd number of transpositions + œ 7" â7#?" 3#@ â3" To show that this is not possible, we choose a representation of + as a product of an odd number of transpositions as follows:

Permutation Groups

195

1) Find the smallest odd integer 7  " for which + is the product of 7 transpositions. 2) Choose an integer 5 that appears in at least one such representation of +. 3) Among all representations of + as a product of 7 transpositions that contain 5 , let 7 be the one whose rightmost appearence of 5 is as far to the left as possible, say 7 œ )" â)>" ÐB 5Ñ)>" â)7 where )>" â)7 does not involve 5 . Note that >   #, since otherwise the only appearance of 5 is in )" and so 7 Á +. However, we can easily move this rightmost occurrence of 5 one transposition to the left by using the following substitutions. Suppose that )>" œ Ð+ ,Ñ. Note that Ð+ ,Ñ Á ÐB 5Ñ since otherwise the two transpositions would cancel, contradicting the definition of 7. 1) If Ð+ ,Ñ and ÐB 5Ñ are disjoint, then they commute Ð+ ,ÑÐB 5Ñ œ ÐB 5ÑÐ+ ,Ñ 2) If Ð+ ,Ñ œ ÐB ,Ñ for , Á 5 , then ÐB ,ÑÐB 5Ñ œ ÐB 5 ,Ñ œ Ð, B 5Ñ œ Ð, 5ÑÐ, BÑ 3) If Ð+ ,Ñ œ Ð5 ,Ñ where + Á 5 , then write Ð5 ,ÑÐB 5Ñ œ Ð5 B ,Ñ œ ÐB , 5Ñ œ ÐB 5ÑÐB ,Ñ Thus, we can move the rightmost occurrence of 5 to the left, contradicting the construction of 7 and proving part 1). For part 2), a cycle of length 7   # can be written as a product of 7  " transpositions as above. Now suppose that the cycle decomposition of 5 is 5 œ -" â-< ." â.= where lenÐ-3 Ñ œ 73   # and lenÐ.3 Ñ œ ". Then 5 can be written as a product of the following number of transpositions: <

Ð73  "Ñ œ Ð8  =Ñ  < œ 8  5

…

3œ"

Generating Sets for W8 and E8 There are a variety of useful generating sets for the symmetric group W8 . For example, we have seen that the set of transpositions generates W8 .

196

Fundamentals of Group Theory

Theorem 6.6 The following sets generate W8 . 1) The set of transpositions. 2) The “transpositions of "” Ð" #Ñß Ð" $Ñß á ß Ð" 8Ñ 3) The “adjacent transpositions” Ð" #Ñß Ð# $Ñß Ð$ %Ñß á ß Ð8  " 8Ñ 4) The 8-cycle Ð"â8Ñ and the transposition Ð" #Ñ. 5) The cycle Ð#â8Ñ and a single transposition Ð" 5Ñ for # Ÿ 5 Ÿ 8. Proof. For part 2) we have Ð+ ,Ñ œ Ð" ,ÑÐ" +Ñ and so the subgroup generated by the transpositions of " contains all transpositions. For part 3), if E is the subgroup generated by the adjacent transpositions, then Ð" #Ñ − E and if Ð" 5Ñ − E, then so is Ð" 5  "Ñ œ Ð" 5ÑÐ5 5"Ñ Hence, part 2) implies that E œ W8 . For part 4), if T œ ØÐ#â8Ñß Ð" #ÑÙ, then Ð" #Ñ − T and if Ð5  " 5Ñ − T , then so is Ð5ß 5  "Ñ œ Ð5  " 5Ñ5 for all 5 Ÿ 8  ". Hence, T œ W8 . For part 5), if 5 œ Ð#â8Ñ, then Ð" 5Ñ5 œ Ð" 5  "Ñ Ð" 5  "Ñ5 œ Ð" 5  #Ñ ã Ð" 8  "Ñ5 œ Ð" 8Ñ and since . œ 5" œ Ð8â#Ñ, Ð" 5Ñ. œ Ð" 5  "Ñ Ð" 5  "Ñ. œ Ð" 5  #Ñ ã Ð" $Ñ. œ Ð" #Ñ and so we get all transpositions of ".… As to the alternating group, we have the following. Theorem 6.7 For 8   $, E8 is generated by the $-cycles. Proof. Any even permutation is a product of pairs of distinct transpositions. For “overlapping” transpositions, we have

Permutation Groups

197

Ð+ ,ÑÐ+ -Ñ œ Ð+ - ,Ñ and for disjoint transpositions, we can arrange for overlaps as follows: Ð+ ,ÑÐ- .Ñ œ Ð+ ,ÑÐ, -ÑÐ, -ÑÐ- .Ñ œ Ð, - +ÑÐ- . ,Ñ Hence, E8 is generated by $-cycles.…

Subgroups of W8 and E8 The alternating group E8 sits very comfortably inside W8 . Theorem 6.8 1) The signum map 9À 5 È Ð"Ñ5 is a group homomorphism from W8 onto the multiplicative group Ö"ß "×, with kernel E8 . Hence, E8 ü W8 has index #. 2) E8 is the only subgroup of W8 of index #. 3) (Subgroups of W8 ) If L Ÿ W8 , then either L Ÿ E8 or L contains an equal number of even and odd permutations and so has even order. 4) (Subgroups of E8 ) For 8   $, E8 has no subgroup of index #. Proof. For part 2), let L be a subgroup of W8 of index #. Theorem 3.17 implies that 5# − L for any 5 − W8 and so the squares of all $-cycles are in L . But any $-cycle is the square of another $-cycle: Ð+ , -Ñ œ Ð+ - ,Ñ# and so L contains all $-cycles, which generate E8 , whence L œ E8 . Part 3) also follows from Theorem 3.17. For part 4), if ÐE8 À LÑ œ #, then 5# − L for all 5 − E8 and so L contains all $-cycles, a contradiction.…

The Alternating Group Is Simple We wish to show that the alternating group E8 is simple for 8 Á %, but not for 8 œ %. We will leave proof of the cases 8 Ÿ % to the reader and assume that 8   &. Suppose that R ü W8 is nontrivial. We have seen that any two $-cycles are conjugate in W8 . However, it is also true that any two $-cycles are conjugate in E8 , for 8   &. To see this let α œ Ð+" +# +$ Ñ and " œ Ð," ,# ,$ Ñ be distinct $cycles with +" Á ,3 for all 3. If 5 œ Ð+" ," ÑÐ+# ,# ÑÐ+$ ,$ Ñ where Ð+3 ,3 Ñ is the identity when +3 œ ,3 , then α5 œ " . Finally, if 5 is not even, then we can take B  Ö+" ß ," ß ,# ß ,$ ×, in which case αÐ+" BÑ5 œ " . Thus, since E8 is generated by $-cycles, we can prove that E8 is simple by showing that any nontrivial normal subgroup of E8 contains at least one $cycle.

198

Fundamentals of Group Theory

E& is Simple To see that E& is simple, note that the possible cycle representations of elements of E& , excluding "-cycles, are Ð † † † † † Ñß

І † †Ñ

and

Ð † † ÑÐ † † Ñ

If R contains a &-cycle, we may assume that 5 œ Ð" # $ % &Ñ − R . To shorten the &-cycle to a $-cycle, we reverse the effects of 5 on " and $ by taking the product 7 œ Ð# " % $ &ÑÐ" # $ % &Ñ œ Ð# & %Ñ and since Ð# " % $ &Ñ œ Ð" # $ % &ÑÐ" #ÑÐ$ %Ñ − R we deduce that R contains the $-cycle 7 . If R contains the product of two disjoint transpositions, we may assume that Ð" #ÑÐ$ %Ñ − R To get a $-cycle, recall that the product of two distinct nondisjoint transpositions is a $-cycle. Thus, Ð" #ÑÐ$ %ÑÐ" #ÑÐ% &Ñ œ Ð$ %ÑÐ% &Ñ œ Ð% & $Ñ and since Ð" #ÑÐ% &Ñ œ ÒÐ" #ÑÐ$ %ÑÓÐ$ % &Ñ − R we see that R contains a $-cycle in this case as well.

E8 is Simple for 8   & We now examine the general case 8   &. Let R ü E8 and let 5 − R . Case 1 Suppose that the cycle decomposition of 5 contains a cycle of length 5   %, say 5 œ 1Ð+" â +7 B C DÑ where 7   ". If - œ ÐB D C +7 â +" Ñ then - and 1 are disjoint and 7 ³ -1" 5 œ ÐB D C +7 â +" ÑÐ+" â +7 B C DÑ œ ÐB +7 DÑ is a $-cycle. Moreover,

Permutation Groups

199

1-" œ 1Ð+" â +7 C D BÑ œ Ò1Ð+" â +7 B C DÑÓÐB C DÑ œ 5ÐB C DÑ − R and so -1" − R , which implies that the $-cycle 7 is in R . Case 2 Suppose that the cycle decomposition of 5 contains two or more $-cycles, say 5 œ Ð+ , -ÑÐB C DÑ1 Then R contains the permutation 7 ³ 5Ð+ C DÑ œ ÐC , -ÑÐB D +Ñ1 and therefore also the &-cycle 57 " œ Ð+ , -ÑÐB C DÑÐ+ D BÑÐ- , CÑ œ Ð+ B , D CÑ Hence, case 1) completes the proof. Case 3 If the cycle decomposition of 5 consists of a single $-cycle, with possibly some additional transpositions, 5 œ Ð+ , -Ñ1" â17 then R contains the $-cycle 5# œ Ð+ , -Ñ# œ Ð+ - ,Ñ Case 4 If the cycle decomposition of 5 is a product of at least three disjoint transpositions, 5 œ Ð+ ,ÑÐB CÑÐD AÑ1" â17 then R also contains 7 ³ 5Ð, BÑÐC DÑ œ Ð+ BÑÐ, DÑÐC AÑ1" â17 and therefore also 57 œ Ð+ ,ÑÐB CÑÐD AÑÐ+ BÑÐ, DÑÐC AÑ œ Ð+ C DÑÐ, A BÑ and so case 2) completes the proof. Case 5 Suppose that the cycle decomposition of 5 has the form 5 œ Ð+ ,ÑÐ- .Ñ If B  Ö+ß ,ß -ß .×, then R contains

200

Fundamentals of Group Theory

7 ³ 5Ð+ , BÑ œ Ð, BÑÐ- .Ñ and therefore also the $-cycle 57 œ Ð+ ,ÑÐ- .ÑÐ, BÑÐ- .Ñ œ Ð+ ,ÑÐ, BÑ œ Ð, B +Ñ Thus, in all cases R contains a $-cycle and so R œ E8 . Theorem 6.9 The alternating group E8 is simple if and only if 8 Á %.… As to the normal subgroups of W8 , we have the following. Theorem 6.10 If 8 Á %, then W8 has no normal subgroups other than Ö"×ß E8 and W8 . Proof. This is easily checked for 8 Ÿ # so assume that 8   $. If R ü W8 , then R ∩ E8 ü E8 . Hence, R ∩ E8 œ Ö+× or R ∩ E8 œ E8 . But if R ∩ E8 œ Ö+×, then R œ Ö+ß 5× where 5 is an odd involution and so has cycle decomposition 5 œ Ð+" ," ÑâÐ+5 ,5 Ñ where the transpositions Ð+3 ,3 Ñ are disjoint. Since R is normal, it must contain all permutations with the same cycle structure as 5, which is not the case if 8   $. Hence, E8 Ÿ R and so R œ E8 or R œ W8 .… Example 6.11 (An infinite simple group) Using the fact that E8 is simple for 8 Á %, we can construct an example of an infinite simple group. Let \ be an infinite set and let W\ be the symmetric group on \ . Thus, 5lsuppÐ5Ñ is a permutation of suppÐ5Ñ with no fixed points and 5 is the identity on \ Ï suppÐ5Ñ. Let WÐ\Ñ be the subgroup of W\ consisting of all elements of W\ that have finite support. Let EÐ\Ñ be the subgroup of WÐ\Ñ consisting of those permutations 5 − WÐ\Ñ for which 5lsuppÐ5Ñ is an even permutation. We leave it as an exercise to show that EÐ\Ñ is an infinite simple group. Moreover, EÐ\Ñ is the smallest nontrivial normal subgroup in W\ .…

Some Counting It is sometimes useful to know how many permutations there are with a particular cycle structure. Theorem 6.12 1) The number of cycles of length 5 in W8 is 8 8x Š ‹Ð5  "Ñx œ 5 5Ð8  5Ñx In particular, the number of 8-cycles in W8 is Ð8  "Ñx.

Permutation Groups

201

2) The number of permutations in W8 whose cycle structure consists of <3 cycles of length 53 , for 3 œ "ß á ß 7 is 8x <7 <" xâ<7 x5"<" â57 3) Let =Ð8ß 5Ñ be the number of permutations in W8 whose cycle structure has exactly 5 cycles (including "-cycles). Then =Ð8ß "Ñ œ Ð8  "Ñx and for 5   # =Ð8ß 5Ñ œ =Ð8  "ß 5  "Ñ  Ð8  "Ñ=Ð8  "ß 5Ñ Also, 8

=Ð8ß 5ÑB5 œ BÐ8Ñ ³ BÐB  "ÑâÐB  8  "Ñ 5œ"

The numbers =Ð8ß 5Ñ are known as the Stirling numbers of the first kind. The expression BÐ8Ñ is known as the 8th upper factorial. Proof. We leave proof of part 1) to the reader. For part 2), we write down a template consisting of <3 cycles of length 53 , with dots in place of the elements of M8 œ Ö"ß á ß 8×. For example, if the cycle lengths are $ß $ß #ß ", then the template is Ð † † † ÑÐ † † † ÑÐ † † ÑÐ † Ñ Now, the dots can be replaced by the elements of M8 in 8x different ways. However, there are two ways in which the number 8x is an overcount of the desired number. First, for each of the <3 cycle templates of length 53 , a cycle can start at any of its 53 elements, so we must divide by 8x by 53<3 , giving 8x <7 5"<" â57 Second, for each cycle length 53 , the <3 x arrangements of the <3 cycles of length 53 are counted separately in the number above, but give the same permutation, so we must also divide by <" xâ<7 x. For part 3), it is easy to see that =Ð8ß "Ñ œ Ð8  "Ñx In general, we group the permutations in W8 with exactly 5 cycles into two groups, depending on whether the permutation contains the "-cycle Ð8Ñ. There are =Ð8  "ß 5  "Ñ permutations containing the "-cycle Ð8Ñ. The other permutations are formed by inserting 8 after any of the 8  " elements in the =Ð8  "ß 5Ñ permutations of 8  " elements into 5 cycles. Thus, for 5   #

202

Fundamentals of Group Theory

=Ð8ß 5Ñ œ =Ð8  "ß 5  "Ñ  Ð8  "Ñ=Ð8  "ß 5Ñ Now we can verify the formula for =Ð8ß 5Ñ by induction. It is easy to see that the formula holds for 8 œ ". Assume the formula is true for =Ð7ß 5Ñ where 7  8. Then 8

=Ð8ß 5ÑB5 5œ" 8

8

=Ð8  "ß 5  "ÑB5  Ð8  "Ñ

œ Ð8  "ÑxB 

5œ# 8"

=Ð8  "ß 5ÑB5 5œ#

=Ð8  "ß 5ÑB5

œ Ð8  "ÑxB  B 5œ"

 Ð8  "Ñ– Ð8"Ñ

œ Ð8  "ÑxB  BÐBÑ œ ÐB  8  "ÑBÐ8"Ñ œ BÐ8Ñ

8 5œ"

=Ð8  "ß 5ÑB5  Ð8  #ÑxB—

 Ð8  "ÑÒBÐ8"Ñ  Ð8  #ÑxBÓ

as desired.…

Exercises 1.

Let 5 œ Ð" # $ÑÐ% &ÑÐ'Ñ and 7 œ Ð% & 'ÑÐ" $ÑÐ#Ñ

2. 3. 4. 5. 6. 7. 8.

be elements of W' . Find a permutation 3 − W' for which 53 œ 7 . Is 3 unique? Find the smallest normal subgroup of W% containing 5 œ Ð" #ÑÐ$ %Ñ. Is this the smallest subgroup of W% containing 5? Let 5 − W8 have prime order :. Prove that 5 is a product of disjoint :cycles. If 5 moves all elements of M8 , show that : ± 8. Show that two even permutations may be conjugate in W& but not in E& . Prove that E8 has no subgroup of index # without using the fact that E8 is simple for 8 Á %. How does this relate to Lagrange's theorem? a) Find all subgroups of E% . b) Find all normal subgroups of E% . Is E% the join of all of its proper normal subgroups? Prove that the $-cycles Ð"#$Ñß Ð"#%Ñß á ß Ð"#8Ñ generate E8 . Let 8   $ and 5   ". Prove that E8 is generated by the cycles of length #5  ".

Permutation Groups 9. 10. 11. 12. 13.

14.

15.

203

Prove that for 8   &, E8 is generated by the set of all products of pairs of disjoint transpositions. Does this hold for 8  &? Let 8   &. Prove that the only proper subgroup L of W8 with ÐW8 À LÑ  8 is E8 . Prove that W8 is centerless for 8   $. Prove that E% is centerless. What about E8 in general? A subgroup L of W8 is transitive if for any 4ß 5 − M8 , there is a 5 − L for which 54 œ 5 . Prove that the order of a transitive subgroup L of W8 is divisible by 8. A subgroup L of W8 is 5 -ply transitive if for any distinct 3" ß á ß 35 − M8 and distinct 4" ß á ß 45 − M8 there is a permutation 5 − L for which 53? œ 4? . Prove that E8 is Ð8  #Ñ-ply transitive. Is it Ð8  "Ñ-ply or 8-ply transitive? Let \ be an infinite set. Define the alternating group E\ is defined to be the set of all permutations in W8 that can be written as the product of an even number of transpositions. Let L\ be the set of all permutations in W\ that fix all but a finite number of elements of \ . a) What is the relationship between E\ , L\ and W\ (including normality and index)? b) Prove that E\ is simple. In W8 , for each 5 œ $ß á ß 8, let ¨ 5# ©

55 œ $ Ð3 5  3Ñ 3œ"

for example, 5$ 5% 5& 5' 5( 5)

œ Ð" #Ñ œ Ð" $Ñ œ Ð" %ÑÐ# $Ñ œ Ð" &ÑÐ# %Ñ œ Ð" 'ÑÐ# &ÑÐ$ %Ñ œ Ð" (ÑÐ# 'ÑÐ$ &Ñ

Show that the permutations 5" ß á ß 58 generate W8" . 16. What is the largest order of the elements of W"! ? 17. (Determining the parity of a permutation) Let \ œ ÖB" ß á ß B8 ×. Let T be the set of all polynomials in the B3 's with rational coefficients. Then we can apply a 5 − W8 to the elements of T by applying 5 to the variables individually. For example, if " : œ B" B$  B$# #

204

Fundamentals of Group Theory then " 5: œ 5ÐB" Ñ5ÐB$ Ñ  5ÐB# Ñ$ − T # a)

Show that for :ß ; − T , 5Ð:  ;Ñ œ 5Ð:Ñ  5Ð;Ñß 5Ð:;Ñ œ 5Ð:Ñ5Ð;Ñ and for 5ß 7 − W\ , 5Ð7 :Ñ œ Ð57 Ñ:

b) Consider the polynomial

: œ :ÐB" ß á ß B8 Ñ œ $ 34

B3  B4 −T 34

Show that if 5 œ ÐB" B+ Ñ is a transposition, then 5: œ $ 34

c)

5ÐB3 Ñ  5ÐB4 Ñ œ : 34

Show that for any 5 − W8 , 5: œ Ð"Ñ5 : Hence, if 5 − W8 , then since :Ð"ß #ß á ß 8Ñ œ ", we have 5:Ð"ß #ß á ß 8Ñ œ $ 34

5Ð3Ñ  5Ð4Ñ œ Ð"Ñ5 34

d) Let 5 − W8 . An inversion in 5 is a pair Ð4ß 3Ñ of indices with " Ÿ 3ß 4 Ÿ 8 satisfying 4  3 but 5Ð4Ñ  5Ð3Ñ. Prove that the sign of 5 is the parity of the number of inversions in 5. e) Determine whether the permutation 5 − W* is even or odd, where 5 œ Ð( # % * $ ) ' & "Ñ 18. For each 7 − W\ , we may associate the conjugation map -7 − W\ defined by -7 Ð5Ñ œ 57 . a) Show that the map -À K Ä W\ defined by -Ð7 Ñ œ -7 is a homomorphism. b) Find the kernel of -. c) Suppose that g Á E © \ is invariant under conjugation, that is, + − E implies that +5 − E. Then we can restrict -5 to a permutation of E. Find the kernel of the map -w À H Ä WE . 19. Let K be any normal subgroup of W8 (such as E8 or W8 itself).

Permutation Groups a)

205

Show that if 5ß 7 ß 3 − K, then 57 œ 53 Í 7 − 3GK Ð5Ñ

b) Find a one-to-one correspondence between the set 5K of conjugates of 5 and the set of cosets of GK Ð5 Ñ in K. The set 5 K is referred to as the conjugacy class of 5 or the orbit of 5 under conjugation. The centralizer GK Ð5Ñ is also referred to as the stabilizer of 5 under conjugation. c) Prove the orbit-stabilizer relationship for conjugation in K: For any 5 − K, ¸5K ¸ œ ÐK À GK Ð5ÑÑ œ

kKk kGK Ð5Ñk

d) Let α − W8 be an 8-cycle. Prove that GW8 ÐαÑ œ ØαÙ. Put another way, a permutation 5 − W8 commutes with α if and only if it is a power of α. 20. If 5ß 7 − E8 are conjugate in W8 , it does not necessarily follow that 5 and 7 are conjugate in E8 . Suppose that 5 − E8 commutes with an odd permutation - − W8 a) Prove that if 5 and 7 are conjugate in W8 , then they are also conjugate in E8 . b) Prove that the centralizers of 5 are related as follows: GW8 Ð5Ñ œ GE8 Ð5Ñ ∪ -GE8 Ð5Ñ and, in particular,

kGW8 Ð5Ñk œ #kGE8 Ð5Ñk

Note: It is not hard to find permutations 5 − E8 that commute with an odd permutation. For instance, if 5 − E8 does not move either + or , , then 5 commutes with the transposition Ð+ ,Ñ. Thus, for instance, an Ð8  #Ñ-cycle in E8 commutes with a transposition. As another example, if 5 − E8 interchanges + and , , then 5 œ .Ð+ ,Ñ.w where . and .w fix + and , . It follows that 5 commutes with Ð+ ,Ñ. 21. Let \ be a nonempty set. Define the support of a permutation 5 − W\ to be the set of elements of \ that are moved by 5: suppÐ5Ñ œ ÖB − \ ± 5B Á B× Let WÐ\Ñ be the set of all permutations in W\ with finite support, that is, for which suppÐ5Ñ is a finite set. This set is called the restricted symmetric group on the set \ and is used in defining the determinant of an infinite matrix. a) Show that suppÐ5Ñ œ suppÐ5" Ñ. b) Show that suppÐ57 Ñ © suppÐ5Ñ ∪ suppÐ7 Ñ. c) Show that suppÐ757 " Ñ œ 7 ÐsuppÐ5ÑÑ. d) Show that suppÐ5Ñ ∩ suppÐ7 Ñ œ g implies that 57 œ 75 .

206

Fundamentals of Group Theory e) f)

Show that WÐ\Ñ ü W\ and that WÐ\Ñ œ W\ if and only if \ is finite. Show that if \ is infinite, then WÐ\Ñ is an infinite group in which every element has finite order and that W\ ÎWÐ\Ñ is infinite. g) Let \ be infinite and let EÐ\Ñ be the subgroup of WÐ\Ñ consisting of those permutations 5 − WÐ\Ñ for which 5lsuppÐ5Ñ is an even permutation. Show that EÐ\Ñ is an infinite simple group. Moreover, EÐ\Ñ is the smallest normal subgroup in W\ . 22. For the Stirling numbers =Ð8ß 5Ñ of the first kind, show that 8

=Ð8ß 5Ñ œ 8x 5œ"

23. (Stirling numbers of the second kind) For the curious, we present a brief discussion of the “other” Stirling numbers. We have seen that the Stirling numbers =Ð8ß 5Ñ of the first kind count the number of ways to partition a set of size 8 into 5 disjoint nonempty “cycles.” The Stirling numbers of the second kind, denoted by WÐ8ß 5Ñ, count the number of ways to partition a set of size 8 into 5 nonempty disjoint subsets. It is clear that WÐ8ß "Ñ œ " a) Find WÐ8ß #Ñ. b) Show that for 8  !, WÐ8ß 5Ñ œ 5WÐ8  "ß 5Ñ  WÐ8  "ß 5  "Ñ c)

Show that 8

B8 œ

WÐ8ß 5ÑBÐ5Ñ 5œ"

Chapter 7

Group Actions; The Structure of :-Groups

Group Actions We have had a few occasions to use group actions -À K Ä W\ in the past and we now wish to make a more systematic study of group actions, beginning with the definition. Definition An action of a group K on a nonempty set \ is a group homomorphism -À K Ä W\ , called the representation map for the action. The permutation -+ is often denoted by -+ , or simply by + itself when no confusion can arise. Thus, "B œ B

and

Ð+,ÑB œ +Ð,BÑ

for all B − \ . When - is a group action, we say that K acts on \ by - or that \ is a K-set under -. An action is faithful if it is an embedding. 1) An element + − K is said to fix B − \ if +B œ B and move B if +B Á B. 2) An element B − \ is stable if every element of K fixes B. We will denote the set of all stable elements by Fix\ ÐKÑ or just FixÐKÑ.… Let \ \ denote the set of all functions from \ to \ . If \ is a finite set, then any function -À K Ä \ \ that satisfies -" œ + and -+, œ -+ -, is an action of K on \, since -+ B œ -+ C Ê -+" -+ B œ -+" -+ C Ê -+" + B œ -+" + C Ê -" B œ -" C ÊBœC and so -+ is injective and therefore a permutation of \ . Thus, for finite sets, we do not need to check explicitly that -+ is a permutation of \ .

S. Roman, Fundamentals of Group Theory: An Advanced Approach, DOI 10.1007/978-0-8176-8301-6_7, © Springer Science+Business Media, LLC 2012

207

208

Fundamentals of Group Theory

Orbits and Stabilizers Two elements Bß C − \ are K-equivalent if there is an + − K for which +B œ C. Since - is a homomorphism, K-equivalence is an equivalence relation on \ . The equivalence classes KB œ Ö+B ± + − K× are called the orbits of the action. We will use the notations KB, orbÐBÑ and orbK ÐBÑ for the orbit of B under K. The distinct orbits form a partition of the set \. On the group side of an action, we can associate to each element B − \ the set of all element of K that fix B: stabÐBÑ œ stabK ÐBÑ œ Ö+ − K ± +B œ B× This subgroup of K is called the stabilizer of B. Definition Let K be a group and \ a nonempty set. 1) An action of K on \ is transitive if every pair of elements of \ are Kequivalent, that is, if there is only one orbit in \ . In this case, we also say that K is transitive on \ . 2) An action of K on \ is regular if it is transitive and if the stabilizer stabÐBÑ is trivial for every B − \ . In this case, we also say that K is regular on \.…

The Kernel of the Representation Map The kernel of the representation map -À K Ä W\ is kerÐ-Ñ œ Ö+ − K ± -+ œ +× As we have remarked, the action - is faithful if kerÐ-Ñ is trivial. Theorem 7.1 The kernel of an action -À K Ä W\ is the intersection of the stabilizers of all elements of \ , kerÐ-Ñ œ , stabÐBÑ

…

B−\

The importance of the kernel O œ kerÐ-Ñ of the representation map -À K Ä W\ stems from two facts: O is a normal subgroup of K and there is an embedding of KÎO into the symmetric group W\ . In particular, if k\ k œ 8 is finite, then ÐK À OÑ ± 8x and if - is faithful, then 9ÐKÑ ± 8x.

The Key Relationships A set consisting of precisely one element from each orbit of K in \ is a system of distinct representatives, or SDR for the orbits in \ .

Group Actions; The Structure of :-Groups

209

For a given SDR, we will have occasion to separate the representatives of the "element orbits from the representatives of the orbits of size greater than ". Accordingly, we denote a system of distinct representatives for the orbits of size greater than " by SDR" . Our immediate goal is to establish some key facts concerning group actions. First, as the various elements of a group K act on an element B − \ , the orbit of B is described. Of course, different elements of K may have the same effect on B. In fact, +B œ ,B

Í

, " + − stabÐBÑ

Í

+stabÐBÑ œ , stabÐBÑ

Thus, +B œ ,B if and only if + and , are in the same coset of stabÐBÑ in K and so we can think of the cosets themselves as acting on the elements B − \ , describing each element of the orbit of B with no duplications, that is, distinct cosets send B − \ to distinct elements of \ . Put another way, there is a bijection between KÎstabÐBÑ and KB. This gives the orbit-stabilizer relationship kKBk œ ÐK À stabÐBÑÑ

It follows that

k\ k œ

B−SDR

kKBk œ

ÐK À stabÐBÑÑ B−SDR

We will refer to this equation as the class equation for the action of K on \ . (Actually, this term is traditionally reserved for a specific case of this equation, arising from the specific action of K on itself by conjugation. We will encounter this specific case a bit later in the chapter.) Another key property of a group action is the following. If Bß C − \ are Kequivalent, say C œ +B for + − K, then the stabilizers of B and C are related as follows: stabÐ+BÑ œ Ö, − K ± ,+B œ +B× œ Ö, − K ± +" ,+B œ B× œ Ö, − K ± +" ,+ − stabÐBÑ× œ stabÐBÑ+ Finally, suppose that K acts on \ and that the restriction of this action to a subgroup L Ÿ K is transitive on \ . Then the action of K is duplicated by the action of L , that is, for any 1 − K and B − \ , there exists an 2 − L for which 2B œ 1B or, equivalently, 1 − 2stabK ÐBÑ. Hence, K œ L stabK ÐBÑ Moreover, if L is regular on \ , then

210

Fundamentals of Group Theory

L ∩ stabK ÐBÑ œ stabL ÐBÑ œ Ö"× and so K œ L ì stabK ÐBÑ Now we summarize. Theorem 7.2 Let the group K act on the set \ . 1) (Orbit-stabilizer relationship) For any B − \ ,

kKBk œ ÐK À stabÐBÑÑ

Thus, for a finite group K,

and kKBk divides kKk. 2) (The class equation)

kKBk œ

k\ k œ

kK k kstabÐBÑk

ÐK À stabÐBÑÑ B−SDR

where the sum is taken over a system of distinct representatives SDR for the orbits in \ . We can also write this as k\ k œ kFix\ ÐKÑk 

ÐK À stabÐBÑÑ B−SDR"

3) (The stabilizer relationship) For any B − \ and + − K, stabÐ+BÑ œ stabÐBÑ+ Thus stabilizers of an orbit in \ form a conjugacy class of K and therefore the stabilizers of K-equivalent elements have the same cardinality. 4) (The Frattini argument) If the action of L Ÿ K is transitive on \ , then K œ L stabK ÐBÑ and if L is regular on \ , then K œ L ì stabK ÐBÑ and so stabK ÐBÑ is a complement of L in K.… When the group action is transitive, the class equation and orbit-stabilizer relationship become quite simple. Theorem 7.3 If a group K acts transitively on a set \ , then all stabilizers are conjugate and the orbit-stabilizer relationship (and the class equation) is simply

Group Actions; The Structure of :-Groups

211

k\ k œ ÐK À stabÐBÑÑ

for any B − \ . Hence, if K is finite, then k\ k divides kKk.…

Congruence Relations on a K -Set If K acts on the elements of a set \ , then there is a natural way in which K also acts on the power set kÐ\Ñ of \ , namely, +W œ Ö+= ± = − W× for all W © \ and + − K. Let us refer to this action as the induced action of K on kÐ\Ñ. Now, a K-set is a set \ with some structure, namely, the group action of K on \ and an equivalence relation ´ on \ is compatible with this action if it satisfies the following definition. Definition An equivalence relation ´ on a K-set \ is called a K-congruence relation on \ if it preserves the group action, that is, if B´C

Ê

+B ´ +C

for all + − K

We denote the set of all congruence classes under ´ by \Î ´ and the congruence class containing B − \ by ÒBÓ.… Thus, if ´ is a K-congruence relation on \ , then the induced action is an action on the partition \Î ´ and +ÒBÓ œ Ò+BÓ for all B − \ . Conversely, suppose that the induced action of K on kÐ\Ñ is an action on a partition c œ ÖF3 ± 3 − M× of \ , that is, +F3 − c for all + − K and F3 − c . Then the equivalence relation ´ associated to c is a K-congruence relation on \ . In other words, the partitions of \ that correspond to the K -congruence relations are the partitions c œ ÖF3 ± 3 − M× that are closed under the induced action of K on kÐ\Ñ. Moreover, if K acts transitively on \ and if ´ is a K -congruence relation on \ , then K also acts transitively on \Î ´ and so \ œ KW ³ Ö+W ± + − K× ´ is the orbit of any given conjugacy class W − \Î ´ . Hence, the partitions of \ that correspond to the K-congruence relations of a transitive group K are the partitions of the form KW , where W © \ is nonempty.

212

Fundamentals of Group Theory

But if KW is a partition of \ , then +W œ W or +W ∩ W œ g for all + − K . Conversely, if +W œ W or +W ∩ W œ g for all + − K, then the distinct members of KW form a partition of \ . To see this, suppose that B − +W ∩ ,W . Then +" B − W ∩ +" ,W and so +" ,W œ W , whence ,W œ +W . Moreover, the transitivity of K implies that KW œ \ . Hence, if W © \ is nonempty, then KW is a partition of \ if and only if +W œ W

or +W ∩ W œ g

for all + − K. Theorem 7.4 Let a group K act on a nonempty set \ . 1) The partitions of \ that correspond to the K-congruence relations on \ are the partitions of \ that are closed under the induced action of K on kÐ\Ñ. 2) Suppose that K acts transitively on \ . a) The partitions of \ that correspond to the K-congruence relations are the partitions of the form KW œ Ö+W ± + − K× where W © \ is nonempty. b) A nonempty subset W of \ is a congruence class under some Kcongruence if and only if W has the property that +W œ W

or +W ∩ W œ g

for all + − K. Such a subset W of \ is called a block of K.… Thus, W © \ is a block of K if and only if KW is a partition of \ . It follows that if W is a block of K , then so is +W for all + − K . Now that we have established the basic properties of group actions, we can examine a few of the most important examples of group actions. Then we will use group actions to explore the structure of :-groups. In the next chapter, we will use group actions to explore the structure of finite groups and to prove the famous Sylow theorems.

Translation by K Earlier in the book, we discussed two fruitful examples of the action of translation by elements of a group K, namely, on the elements of K and on the elements of a quotient set KÎL . The action of K on itself is the left regular representation of K as a subgroup of the symmetric group WK , as described by Cayley's theorem.

Group Actions; The Structure of :-Groups

213

Let us review the action of translation by K on KÎL : -1 Ð+LÑ œ 1+L This action is transitive and so all stabilizers are conjugate. Since stabÐ+LÑ œ stabÐLÑ+ œ L + the kernel of the action is the normal closure of L ,

kerÐ-Ñ œ , L + œ L ‰ +−K

which is the largest normal subgroup of K contained in L . If ÐK À LÑ œ 7, then the embedding K ä WÐKÀLÑ ¸ W7 L‰ implies that ÐK À L ‰ Ñ ± ÐK À LÑx The consequences of this action were recorded earlier in Theorem 4.20, but we repeat them here for easy reference. Theorem 7.5 Let K be a group and let L  K have finite index. Then KÎL ‰ ä WKÎL and so ÐK À L ‰ Ñ ± ÐK À LÑx In particular, ÐK À L ‰ Ñ is also finite and ÐL À L ‰ Ñ ± ÐÐK À LÑ  "Ñx 1) Any of the following imply that L ü K: a) L is periodic and ÐK À LÑ œ : is equal to the smallest order among the nonidentity elements of L . b) K is finite and 9ÐLÑ and ÐÐK À LÑ  "Ñx are relatively prime, that is, for all primes :, : ± 9ÐLÑ

Ê

:   ÐK À LÑ

This happens, in particular, if ÐK À LÑ is the smallest prime dividing 9ÐKÑ. 2) If K is finitely generated, then K has at most a finite number of subgroups of any finite index 7.

214

Fundamentals of Group Theory

3) If K is simple, then 9ÐKÑ ± ÐK À LÑx a) If K is infinite, then K has no proper subgroups of finite index. b) If K is finite and 9ÐKÑ y± 7x for some integer 7, then K has no subgroups of index 7 or less.…

Conjugation by K on the Conjugates of a Subgroup The elements of K act by conjugation on subÐKÑ, -+ ÐLÑ œ L + for all L Ÿ K. The orbit of a subgroup L is the conjugacy class conjK ÐLÑ and the stable elements are FixÐKÑ œ norÐKÑ The stabilizer of a subgroup L is its normalizer and so RK ÐL + Ñ œ RK ÐLÑ+ Also, the orbit-stabilizer relationship is

kconjK ÐLÑk œ ÐK À RK ÐLÑÑ

which we discussed earlier in the book (Theorem 3.27).

Conjugation by K on a Normal Subgroup Let R ü K and let K act on the elements of R by conjugation: -1 Ð+Ñ œ +1 for all + − R . The orbits of this action K+ œ +K ³ Ö+B ± B − K× are called the conjugacy classes of R under K. The stabilizer of + − R is its centralizer GK Ð+Ñ and the kernel of the representation - is kerÐ-Ñ œ , GK Ð+Ñ œ GK ÐR Ñ +−R

The orbit-stabilizer relationship is

¸+K ¸ œ ÐK À GK Ð+ÑÑ

as we saw in Theorem 3.23. The stable elements of R are the elements of R that commute with every element of K and so

Group Actions; The Structure of :-Groups

215

FixR ÐKÑ œ ^ÐKÑ ∩ R Hence, the class equation is

kR k œ k^ÐKÑ ∩ R k 

ÐK À GK Ð+ÑÑ +−SDR"

When K acts on itself by conjugation, that is, when R œ K, the class equation is kKk œ k^ÐKÑk 

ÐK À GK Ð+ÑÑ +−SDR"

This is the equation to which the name class equation is traditionally applied and is one of the most useful tools in finite group theory.

The Structure of Finite : -Groups We now wish to study the structure of a very special type of finite group. Definition Let K be a nontrivial group and let : be a prime. 1) An element + − K is called a :-element if 9Ð+Ñ œ :5 for some 5   !. 2) K is a :-group if every element of K is a :-element. 3) A nontrivial subgroup W of K is called a :-subgroup of K if W is a :group.… As we saw earlier in the book, Lagrange's theorem and Cauchy's theorem conspire to give the following result. Theorem 7.6 A finite group K is a :-group if and only if the order of K is a power of :.… When a :-group K acts on a set \ , the class equation has the property that all of the terms ÐK À stabÐBÑÑ that are greater than " are divisible by :. This gives the following simple but useful result. Theorem 7.7 If a :-group K acts on a set \ , then

k\ k ´ kFix\ ÐKÑk mod :

…

We now turn to the key properties of finite :-groups.

The Center-Intersection Property It will be convenient to make the following nonstandard definition. Definition A group K has the center-intersection property if every nontrivial normal subgroup of K intersects the center of K nontrivially.…

216

Fundamentals of Group Theory

Note that a finite group K has the center-intersection property if and only if every nontrivial normal subgroup of K contains a central subgroup of prime order. Any finite :-group K has the center-intersection property, for if R ü K is nontrivial, then K acts on the elements of R by conjugation and Theorem 7.7 implies that kR k ´ k^ÐKÑ ∩ R k mod :

which shows that k^ÐKÑ ∩ R k  ".

Theorem 7.8 A finite :-group K has the center-intersection property. 1) ^ÐKÑ is nontrivial. 2) K is simple if and only if 9ÐKÑ œ :.… The fact that the center of a :-group is nontrivial tells us something very significant about groups of order :# . Corollary 7.9 If 9ÐKÑ œ :# , then K is abelian. In fact, K is either cyclic or is the direct product of two cyclic subgroups of order :. Proof. We must have k^ÐKÑk œ : or :# . But if k^ÐKÑk œ :, then KÎ^ÐKÑ is cyclic and so K is abelian, which is a contradiction. Hence, k^ÐKÑk œ : # and K is abelian. If K is not cyclic, then K is elementary abelian of exponent : and so is the direct product of two cyclic subgroups of order :.…

:-Series and Nilpotence We next show that :-groups have normal subgroups of all possible orders. But first a couple of definitions. Definition (Central Series and :-Series) Let K be a group. 1) A normal series L! ü L" ü â ü L8 in K is central in K if each factor group L5" ÎL5 is central in KÎL5 , that is, L5" ÎL5 Ÿ ^ÐKÎL5 Ñ A group is nilpotent if it has a central series starting at the trivial subgroup Ö"× and ending at K. 2) If : is a prime, then a :-series from L to K is a series L œ L! – L" – â – L8 œ K whose steps L5  L5" have index :.… Definition Let : be a prime. If L  O is an extension in K of index :, we refer to O as a :-cover of L . (O is a cover of L in the lattice subÐKÑ.)…

Group Actions; The Structure of :-Groups

217

Theorem 7.5 implies that if O is a :-cover of L , then L – O . Theorem 7.10 Let K be a finite :-group. Then every L  K has a :-cover O and if L – K, then O can be chosen so that L – O is central in K. 1) There is a :-series from L to K. 2) If L ü K, then there is a central :-series from L to K. In particular, K has a normal subgroup containing L of every order between 9ÐLÑ and 9ÐKÑ (under division). 3) K is nilpotent. Proof. The proof is by induction on 9ÐKÑ. The theorem is true if 9ÐKÑ œ :. Assume 9ÐKÑ  : and that the theorem is true for all groups smaller than K . Let L  K and let R be central in K of order : . If L œ Ö"×, then R is a : -cover of L and L – R is central in K . Assume that L Á Ö"×. We may also assume that R Ÿ L if L – K. In any case, R Ÿ L or R ∩ L œ Ö"×. If R ∩ L œ Ö"×, then R L is a :-cover of L and if R Ÿ L , then the induction hypothesis implies that LÎR has a :-cover OÎR and so O is a :-cover of L . Also, if L – K, the inductive hypothesis implies that LÎR – OÎR is central in KÎR for some O Ÿ K and so Theorem 4.11 implies that L – O is central in K.…

The Normalizer Condition Theorem 7.10 implies that a : -group has the normalizer condition, that is, L K

Ê

L  RK ÐLÑ

and therefore several other nice properties (see the discussion following Theorem 4.35). Corollary 7.11 The following hold in a finite :-group K: 1) K has the normalizer condition 2) Every subgroup of K is subnormal 3) Every maximal subgroup of K is normal 4) KÎFÐKÑ is abelian.…

Maximal and Minimal Subgroups Maximal and minimal subgroups play a key role in the study of finite :-groups. For a finite :-group K, Cauchy's theorem implies that a subgroup L is minimal if and only if 9ÐLÑ œ : and the center-intersection property implies that L is minimal normal if and only if it is central of order :. As to the maximal subgroups of K, Theorem 7.10 implies that a subgroup L is maximal in K if and only L has index : and that L is maximal normal in K if and only if L has index :.

218

Fundamentals of Group Theory

Theorem 7.12 (Maximal and minimal subgroups) Let K be a finite :-group and let L Ÿ K. 1) L is minimal if and only if it has order :. 2) L is minimal normal if and only if it is central of order :. 3) The following are equivalent: a) L is maximal b) L is maximal normal c) ÐK À LÑ œ :.…

The Frattini Subgroup of a :-Group; The Burnside Basis Theorem The fact that any maximal subgroup Q of a :-group K is normal and has index : implies that if + − K, then Ð+Q Ñ: œ Q and so +: − Q . Thus, K: © FÐKÑ, the Frattini subgroup of K. It follows that KÎFÐKÑ is an elementary abelian group of exponent :. Conversely, if KÎO is elementary abelian, then it is characteristically simple and so FÐKÎOÑ œ ÖO×. Hence, FÐKÑ Ÿ , Q Ÿ O OŸQ K Q maximal

We have shown that FÐKÑ is the smallest normal subgroup O of K for which the quotient KÎO is elementary abelian. Theorem 7.13 Let : be a prime. Let K be a group of order :8 , with Frattini subgroup FÐKÑ of order :7 . 1) FÐKÑ is the smallest normal subgroup of K for which KÎFÐKÑ is an elementary abelian group. Moreover, KÎFÐKÑ has exponent : and so is a vector space over ™: , of dimension 8  7. 2) FÐKÑ œ Kw K: 3) (The Burnside Basis Theorem) Any generating set for K contains a generating set of size 8  7. Proof. Part 1) has been proved. For part 2), since ÖKw +: ± + − K× is a subgroup of KÎKw , it follows that Kw K: is a normal subgroup of K. In fact, KÎKw K: is elementary abelian of exponent : and so part 1) implies that FÐKÑ Ÿ Kw K: Ÿ FÐKÑ. Hence, FÐKÑ œ Kw K: . For part 3), write F œ FÐKÑ. We show that K œ Ø1" ß á ß 15 Ù

Í

KÎF œ Ø1" Fß á ß 15 FÙ

One direction is clear and since F is the set of nongenerators of K , we have KÎF œ Ø1" Fß á ß 15 FÙ

Ê

K œ ØÖ1" ß á ß 15 × ∪ FÙ œ Ø1" ß á ß 15 Ù

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219

Thus, since KÎF is a ™: -space of dimension 8  7, any generating set for KÎF contains a generating set of size 8  7. Hence, the same holds true for K.…

Number of Subgroups of a :-Group We now wish to inquire about the number of subgroups of a given size :. in a :-group K of order :8. Let sub. ÐKÑ and nor. ÐKÑ denote the families of subgroups and normal subgroups, respectively, of K of size :. . Then K acts by conjugation on sub. ÐKÑ and the stable set is nor. ÐKÑ, whence ksub. ÐKÑk ´ knor. ÐKÑk mod :

Our plan is to show that knor. ÐKÑk ´ " mod :.

Theorem 7.14 Let K be a nontrivial :-group of order :8 . 1) The number of maximal subgroups of K is ksub8" ÐKÑk œ knor8" ÐKÑk œ 2) For any ! Ÿ . Ÿ 8,

:87  " ´ " mod : :"

ksub. ÐKÑk ´ knor. ÐKÑk ´ " mod :

Proof. For part 1), if 9ÐFÐKÑÑ œ :7 , then Theorem 7.13 implies that E œ KÎFÐKÑ is a vector space over ™: of dimension 8  7. In general, if Z is a vector space over ™: of dimension 5 , then the number of subspaces of Z of dimension . is Z Ð5ß .Ñ œ

Ð:5  "ÑÐ:5  :ÑâÐ:5  :." Ñ Ð:.  "ÑÐ:.  :ÑâÐ:.  :." Ñ

(We ask the reader to supply a proof in the exercises.) Hence, the number of subgroups (subspaces) of Z of order :5" is Z Ð5ß 5  "Ñ œ

Ð:5  "ÑÐ:5  :ÑâÐ:5  :5# Ñ :5  " œ Ð:5"  "ÑÐ:5"  :ÑâÐ:5"  :5# Ñ :"

In particular, the number of maximal subgroups of KÎFÐKÑ is Z Ð8  7ß 8  7  "Ñ œ

:87  " :"

and this is also the number of maximal subgroups of K. For part 2), let ?. ÐKÑ œ knor. ÐKÑk and let ´ stand for congruence modulo : . Then part 1) implies that ??" ÐKÑ ´ ". We show that ?. ÐKÑ ´ " by induction on 9ÐKÑ. If 9ÐKÑ œ :, the result is clear. Assume it is true for :-groups smaller than K.

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If Q − nor8" ÐKÑ, then K acts on sub. ÐQ Ñ by conjugation. The stable set is nor. ÐKÑ ∩ sub. ÐQ Ñ and so the inductive hypothesis implies that knor. ÐKÑ ∩ sub. ÐQ Ñk ´ ksub. ÐQ Ñk ´ "

Hence, the set f œ ÖÐR ß Q Ñ ± R  Q ß R − nor. ÐKÑß Q − nor8" ÐKÑ×

has size kf k ´ ?8" ÐKÑ ´ ".

On the other hand, for each R − nor. ÐKÑ, there is one Q − nor8" ÐKÑ containing R for each maximal subgroup of KÎR and since ?8." ÐKÎR Ñ ´ ", we have kf k ´ ?. ÐKÑ. Thus, ?. ÐKÑ ´ ".…

*Conjugates in a :-Group

In the study of finite :-groups, it can be useful to examine the conjugates of an element + by the powers of another element , . Theorem 7.15 Let K be a finite :-group and let + − K have order 9Ð+Ñ œ :7 . Let , − K and suppose that +, œ +α for some integer α ´ y " mod :7 . Let +Ø,Ù be the set of conjugates of + by the elements of Ø,Ù. Then ¸+Ø,Ù ¸ œ :.

.

where .   " and :. is the smallest power of : for which , : commutes with +. 1) If :  # or if α ´ y $ mod %, then +Ø,Ù œ Ö+"5:

7.

± 5 œ "ß á ß :. ×

where 7  .   " and +"  +Ø,Ù . 2) If : œ # and α ´ $ mod %, then one of the following holds: a) If .   ", then +Ø,Ù œ Ö+/5 5:

7.

± 5 œ "ß á ß :. ×

where 7  .   ", +"  +Ø,Ù and half of the /5 's are " and half are ". b) If . œ !, then +Ø,Ù œ Ö+ß +"#

7"

×

or +Ø,Ù œ Ö+ß +" ×

3) a) If :  #, then no element of K of order : can be conjugate to one of its own powers other than the first power.

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221

b) If : œ #, then no element + − K of order # can be conjugate to one of its own powers other than + or +" . Proof. The conjugates of + by Ø,Ù are , 5 +, 5 œ +α

5

for 5 œ "ß á ß <, where < œ ¸+Ø,Ù ¸. In fact, < is the smallest positive integer for which , < commutes with +. Also, +Ø,Ù is the orbit of + under conjugation by Ø,Ù and so < œ :. for some .   ". <

Note that < is also the smallest positive integer for which +α œ +, that is, the smallest positive integer for which α< ´ " mod :7 and so .

3) : 7 ± α :  " ." 4) :7 y± α:  " .

Furthermore, since α: ´ " mod :, Fermat's little theorem implies that α ´ " mod :. Thus, if α œ /  -:> is in :-standard form, then Lemma 1.18 implies that for any ?   !, ?

?

α: œ /:  A:?> where : y± A. In particular, ."

."

5) :.>" ± α:  /: . . . 6) :.>" y± α:  /: œ α:  " From 3) and 6), we see that 7 Ÿ .  >. ."

Now, if /: œ " then 4) and 5) imply that .  > Ÿ 7 and so 7 œ >  . . This implies that 7  . œ >   ". Also, α œ /  -:7. and so for " Ÿ 5 Ÿ :. , α5 œ Ð/  -:7. Ñ5 œ /5  :7. A5 Hence, 5

5

+α œ + /

:7. A5

where no two distinct A5 's are congruent modulo :. , since otherwise we would not get :. distinct conjugates. Thus, we can assume that A5 ranges over the set Ö"ß á ß :. × and so +Ø,Ù œ Ö+/5 5:

7.

± 5 œ "ß á ß :. ×

Now, if :  # or α ´ y $ mod %, then / œ " and so /5 œ " for all 5 . If : œ # and

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.

α ´ $ mod % but .  ", then we still have /: œ " but since / œ ", as 5 ranges from " to :. , the term /5 alternates between " and " and so half of the terms /5 are " and half are ". Also, as 5 ranges from " to :. , the exponents /5  5:7. range from „"  : to „"  :7 . But + is conjugate to itself and so one of these exponents must be conjugate to " modulo :7 . Therefore, the last exponent is "  :7 and since no other exponent is conjugate to " modulo :7 , it follows that +"  +Ø,Ù . ."

The case /: œ " occurs precisely when : œ #, α ´ $ mod % and . œ ", in which case + has exactly two conjugates and α œ "  -#> with - odd and >   #. >

If >   7, then +α œ +"-# œ +" and so +Ø,Ù œ Ö+ß +" ×. If >  7, then 7 Ÿ "  > implies that > œ 7  " and so α œ "  -#7" where - is odd, that is, - œ ". Hence, the two conjugates of + are + itself and +α œ +"#

7"

Note finally that the case +Ø,Ù œ Ö+ß +" × does occur in the dihedral group H#7" œ Ø3ß 5Ù where 9Ð3Ñ œ #7 and 9Ð5Ñ œ # and 35 œ 3" . Also, the case +Ø,Ù œ Ö+ß +"#

7"

×

occurs in the semidihedral group WH7 œ Øαß 0Ùß

9ÐαÑ œ #7 ß

9Ð0Ñ œ #ß

0α œ α#

7"

"

0

For part 3), if 9Ð+Ñ œ :, then the number of conjugates of + is both a power of : and at most :  ", whence it must equal ".…

*Unique Subgroups in a :-Group A cyclic :-group has a unique subgroup of order :. If :  #, then the converse of this is true: A :-group that has a unique subgroup of order : is cyclic. We begin with a definition. Definition A generalized quaternion group of order #8 , 8   # is a group U8 with the following properties: U8 œ Ø+ß ,Ùß 9Ð+Ñ œ #8" ß 9Ð,Ñ œ %ß , # œ +#

8#

ß ,+, " œ +"

If 8 œ $, then U8 is a quaternion group.… We will show later in the book that such a group exists: It is a special case of the dicyclic group. We leave it as an exercise to show that Ø,# Ù is the only subgroup of order # in U7 but that for any #  #=  #8 , the group U8 has at least two subgroups of order #= . Also, any B − U8 Ï Ø+Ù has the form B œ +5 , , where

Group Actions; The Structure of :-Groups

Ð+5 ,Ñ# œ +5 Ð,+5 Ñ, œ , # œ +#

223

7"

and so 9Ð+5 ,Ñ œ %. Thus, any element of U8 Ï Ø+Ù has order %. It follows that if 8   %, then Ø+Ù is the unique cyclic subgroup of U8 of order #8" and so Ø+Ù « U8 . We will prove that if a : -group K has a unique subgroup of order :, then K is cyclic if :  # and K is either cyclic or generalized quaternion if : œ #. First, let us show that if K has a unique subgroup L= of any order := , where : Ÿ :=  9ÐKÑ, then K must have a unique subgroup of order :. Since a :-group has subgroups of all orders dividing the order of the group, we have for any subgroup O Ÿ K , 9ÐL= Ñ Ÿ 9ÐOÑ

Ê

L= Ÿ O

Also, since any subgroup O Ÿ K of order less than := is contained in some subgroup of order := , we have 9ÐOÑ Ÿ 9ÐL= Ñ

Ê

O Ÿ L=

Thus, all subgroups of K either contain L= or are contained in L= . In this sense, L= forms a bottleneck in the lattice of subgroups of K. It follows that L= is cyclic, for if + Â L= , then Ø+Ù Ÿ y L= and so L= Ÿ Ø+Ù, whence L= is cyclic. Since L= is cyclic, the subgroup lattice of K has the form shown in Figure 7.1 and so K contains exactly one subgroup of each order :. with ! Ÿ . Ÿ =. In particular, K has a unique subgroup of order :. Thus, we have shown that K has a unique subgroup of some order := , where : Ÿ :=  9ÐKÑ if and only if K has a unique subgroup of order :.

G ...

...

Hs

{1} Figure 7.1

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Fundamentals of Group Theory

Let K be a noncyclic group of order :8  ". To show that K has more than one subgroup of order :, it suffices to show K contains a nontrivial subgroup E as well as an element of order : that is not in E. We first consider the case :  #. Theorem 7.16 Let :  # be prime and let 9ÐKÑ œ :8  ". 1) If K is noncyclic and if + − K is an element of maximum order, then K has an element of order : that is not contained in Ø+Ù. 2) If K has a unique subgroup of order := for some " Ÿ =  8, then K is cyclic. Proof. We have already seen that part 2) follows from part 1). To prove part 1), assume that K is noncyclic. Let E œ Ø+Ù, where + − K has maximum order :7 . If 7 œ ", then all nonidentity elements of K have order : and so we may assume that # Ÿ 7  8. Let E – F where 9ÐFÑ œ :7" . If , − F Ï E, then , : − E and so , : œ +> for some >   !. If > œ !, then 9Ð,Ñ œ : and we are done, so let us assume that >  !. Since +> œ , : does not have maximum order in K , it follows that : ± > and so , : œ +?: for some !  ?  :7" . Now, if , commutes with +, then ,+?  E has order :. Assume now that no , − F Ï E commutes with +. We wish to show that there is still an integer B for which 9Ð,+B Ñ œ :. If , − F Ï E, then to get a formula for Ð,+B Ñ: , we need a commutativity rule for + and , . But since E – F , there is an α Á " for which +, œ +α and since , : commutes with +, Theorem 7.15 implies that +Ø,Ù œ Ö+"5:

7"

± 5 œ "ß á ß :×

Moreover, ,  E implies that , 3 − F Ï E for all " Ÿ 3  : and so we may replace , by an appropriate power of , so that α œ "  : 7" . Now, ,+ œ +α , and so ,+B œ +αB , Then an easy induction shows that for 5   ", #

5

Ð,+B Ñ5 œ +BÐαα âα Ñ , 5 and so #

:

#

:

Ð,+B Ñ: œ +BÐαα âα Ñ , : œ +?:BÐαα âα Ñ Hence, we want an integer B for which

Group Actions; The Structure of :-Groups

Bα

"  α: ´ ?: "α

225

Ðmod :7 Ñ

Since α œ "  :7" , Theorem 1.18 implies that α: œ Ð"  :7" Ñ: œ "  A:7 where : y± A and so Bα

"  α: œ BÐ"  :7" ÑA: ´ BA: "α

Ðmod :7 Ñ

Hence, we want an integer B for which BA ´ ?

Ðmod :7" Ñ

But A is invertible in ™‡:7" and so we may take B œ ?A" .… Now we turn to the case : œ #. (The reader may wish to skip the proof upon first reading.) Theorem 7.17 Let K be a nontrivial group of order #8 . 1) K has a unique subgroup of order # if and only if K is cyclic or a generalized quaternion group. 2) If K has a unique subgroup of order #= for some "  =  8, then K is cyclic. Proof. We have already seen that part 2) follows from part 1). Let K be a noncyclic group of order #8 with a unique involution. We will show that K is a generalized quaternion group. Let + − K be an element of maximum order #7 and let E œ Ø+Ù. Clearly, we may assume that 7   #. In one case we will need to be a bit more specific about the choice of the cyclic subgroup E. Namely, if 7 œ #, then since the unique subgroup of order # is normal in K, it has a #-cover R of order %, which is cyclic since the %-group has two involutions. In this case, we let E œ R and so E ü K. Thus, if 7 œ #, we may assume that E – K. If F œ Ø,ß EÙ is a #-cover of E, then 9ÐFÎEÑ œ # and so , # − E , which implies that , # œ +5 for some 5  !. But 9Ð+5 Ñ œ 9Ð, # Ñ  9Ð,Ñ Ÿ 9Ð+Ñ and so # ± 5 , that is, >

, # œ +# ? for >   " and ? odd. Since Ø+? Ù œ E, we can rename +? to + to get , # œ +# >

>

for >   ". Since 9Ð, # Ñ œ 9Ð+# Ñ œ #7> , we also have

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Fundamentals of Group Theory

9Ð,Ñ œ #7>" where > Ÿ 7  " since 9Ð,Ñ  #. >"

If , commutes with +, then ,+#  E is an involution, contrary to assumption. Thus, , does not commute with +; in fact, E is not properly contained in any abelian subgroup of K. However, since E – F and , # − E, Theorem 7.15 implies that +Ø,Ù œ Ö+ß +"#

7"

or +Ø,Ù œ Ö+ß +" ×

×

and so for any #-cover F œ Ø,ß EÙ of E, there is an + − E for which ,+, " œ +α

>

and , # œ +#

where either α œ "  #7" or α œ ". Conjugating the second equation by , and using the first equation gives >

>

,# œ ,+# , " œ +α# œ +# >

>

>

>+1

and so +# œ +# , that is, +# œ ". Hence, #7 ± #>" , which implies that 7 Ÿ >  " Ÿ 7, that is, > œ 7  ". Now, if for any #-cover F œ Ø,ß EÙ, we have α œ "  #7" , then α œ "  #> and so >

,+, " œ +"# œ , # +" which implies that +, "  E is an involution, a contradiction. Hence, for all #covers F œ Ø,ß EÙ of E, we have α œ " and ,+, " œ +"

and

7"

, # œ +#

It follows that 9Ð,Ñ œ % and so F can be described as follows: F œ Ø+ß ,Ùß 9Ð+Ñ œ #7 ß 9Ð,Ñ œ %ß , # œ +#

7"

ß ,+, " œ +"

that is, F œ U7" is generalized quaternion and so every element of F Ï E has order % and if 7   $, then E « F . We want to show that F œ K. If not, then F has a #-cover G , that is, E–F–G ŸK where 9ÐGÑ œ :72 . Recall that if 7 œ #, then we have chosen E so that E ü K and if 7   $, then E « F and so in either case, E – G .

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227

The quotient group GÎE is either G% Ð-EÑ or G# Ð-EÑ  G# Ð.EÑ. In the latter case, the subgroups Ø-ß EÙ and Ø.ß EÙ are #-covers of E and so +- œ +" œ +. Hence, +-. œ + which implies that E  ØEß -.Ù is abelian, a contradiction. Hence, GÎE œ G% Ð-EÑ. But then Ø- # ß EÙ is a #-cover of E and so #

+- œ +" However, since E is not properly contained in an abelian subgroup of K, the smallest power of - that commutes with + is - % − E and so Theorem 7.0 (where . œ #) implies that +"  +Ø-Ù , a contradiction. Thus K œ F œ U7" is generalized quaternion of order #7 .…

*Groups of Order : 8 With an Element of Order :8" We can use the previous result to take a close look at nonabelian groups K of order :8 that have an element of order : 8" . We will restrict attention to the case :  #. Let 9Ð+Ñ œ :8" and E œ Ø+Ù. Theorem 7.16 implies that there is a , − K Ï E with 9Ð,Ñ œ :. Then K œ Ø+Ù z Ø,Ù and it remains to see how + and , interact. Since E ü K, we have , " +, œ +5 for some 5  " and Theorem 7.15 implies that the conjugates of + by Ø,Ù are +Ø,Ù œ Ö+"5:

8#

± 5 œ "ß á ß :×

Since any nonidentity element of Ø,Ù generates Ø,Ù, we can take 5 œ " and write ,+, " œ +":

8#

Theorem 7.18 Let :  # be a prime. Let K be a nonabelian group of order :8 with an element + of order :8" . Then K œ Ø+Ù z Ø,Ù where 9Ð,Ñ œ : and ,+, " œ +":

8#

…

To see that such a group exists, recall from Example 5.30 that there is a semidirect product

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Fundamentals of Group Theory

G:8" Ð+Ñ z ) G: Ð,Ñ where ), Ð+Ñ œ +":

8#

…

*Groups of Order : $ We have seen that groups of order : are cyclic and that groups of order :# are either cyclic or the direct product of two cyclic subgroups of order : (Corollary 7.9). Theorem 7.18 gives us insight into groups of order :$ . If : œ #, we have seen that, up to isomorphism, the groups of order :$ œ ) are 1) 2) 3) 4) 5)

G) G% } G# G# } G# } G# U, the (nonabelian) quaternion group H) , the (nonabelian) dihedral group

More generally, we will show that for any prime :, the groups of order :$ are (up to isomorphism): 1) 2) 3) 4) 5)

G:$ G:# } G: G: } G: } G: Y X Ð$ß ™: Ñ, the unitriangular matrix group (described below) The group K œ Ø+ß ,Ù where K œ Ø+ß ,Ùß 9Ð+Ñ œ : # ß 9Ð,Ñ œ :ß ,+, " œ +":

Thus, there are only two nonabelian groups of order :$ (up to isomorphism). We will leave analysis of the abelian groups of order :$ to a later chapter, where we will prove that any finite abelian group is the direct product of cyclic groups. So let :  # be prime and let K be a nonabelian group of order :$ . If K has an element + of order :# , then Theorem 7.18 implies that K œ Ø+Ù z Ø,Ù where 9Ð,Ñ œ : and , " +, œ +": It remains to consider the case where K has exponent :. The center ^ œ ^ÐKÑ is nontrivial but cannot have order :# , since then KÎ^ is cyclic and so K is abelian. Hence, 9Ð^Ñ œ : and so 9ÐKÎ^Ñ œ :# . Hence, KÎ^ is abelian with exponent : and so

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229

KÎ^ œ Ø+^Ù  Ø,^Ù Moreover, since D ³ Ò,ß +Ó − ^ , we have ^ œ ØDÙ. Hence, K œ Ø+ß ,ß DÙß 9Ð+Ñ œ 9Ð,Ñ œ 9ÐDÑ œ "ß D − ^ÐKÑß ,+ œ D+, To see that this does describe a group, we have the following. Definition Let V be a commutative ring with identity. A matrix Q − KPÐ8ß VÑ is unitriangular if it is upper triangular (has !'s below the main diagonal) and has "'s on the main diagonal. We denote the set of all unitriangular matrices by Y X Ð8ß VÑ.… We will leave it as an exercise to show that

kY X Ð8ß ™: Ñk œ :Ð8 8ÑÎ# #

and that for :  #, the group Y X Ð$ß ™: Ñ has order :$ and exponent :. Also, Y X Ð$ß ™# Ñ ¸ U.

Exercises 1.

2.

3. 4. 5. 6.

A left action of K on \ is sometimes defined as a map from the cartesian product K ‚ \ to \ , sending Ð+ß BÑ to an element +B − \ , satisfying a) "B œ B for all B − \ b) Ð+,ÑB œ +Ð,BÑ for all B − \ , +, , − K . A right action of K on \ is a map from the cartesian product \ ‚ K to \ , sending ÐBß +Ñ to an element B+ − \ , satisfying c) B" œ B for all B − \ d) BÐ+,Ñ œ ÐB+Ñ, for all B − \ , +, , − K . Given a left action, show that the map ÐBß 1Ñ œ 1" B is a right action. What about ÐBß 1Ñ œ 1B? Let -À K Ä W\ be an action of K on \ . a) Prove that - is regular if and only if it is transitive and stabÐBÑ œ Ö"× for some B − \ . b) Prove that - is regular if and only if it is transitive and for all distinct 1ß 2 − K, we have 1B Á 2B for all B − \ . c) Prove that if - is faithful and transitive and if K is abelian, then the action is regular. Let K be a finite group and let : be the smallest prime dividing 9ÐKÑ. Prove that any normal subgroup of order : is central. Let K be an infinite group. Use normal interiors (not Poincaré's theorem) to prove that if L and O have finite index in K, then so does L ∩ O . Show that the condition that K be finitely generated cannot be removed from the hypotheses of Theorem 7.5. Let K be a finite simple group and let L Ÿ K have prime index ÐK À LÑ œ :. Prove that : must be the largest prime dividing 9ÐKÑ and that :# does not divide 9ÐKÑ.

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Let K be a finite group. Prove that a transitive action of K on \ is regular if and only if kKk œ k\ k. 8. Let 9ÐKÑ œ #8 where 8   " is odd. Let + − K have order #. Show that under the left regular representation of K on itself, the element + corresponds to an odd permutation. Show that K is not simple. 9. a) Prove that if K is a finitely generated infinite group and L is a subgroup of finite index in K, then K has a characteristic subgroup O of finite index for which O Ÿ L . b) Show that the condition that K be finitely generated is necessary. 10. The action of a group K on a set \ is #-transitive if for any pairs ÐBß CÑß Ð?ß @Ñ − \ ‚ \ where B Á C and ? Á @, there is an + − K for which +B œ ? and +C œ @. Prove that for a #-transitive action, the stabilizer stabÐBÑ is a maximal subgroup of K for all B − \ . 7.

Equivalence of Actions Two group actions -À K Ä W\ and .À L Ä W] are equivalent if there is a pair Ðαß 0 Ñ where αÀ K Ä L is a group isomorphism and 0 À \ Ä ] is a bijection satisfying the condition 0 Ð1BÑ œ Ðα1ÑÐ0 BÑ In this case, we refer to Ðαß 0 Ñ as an equivalence from - to .. 11. a) Show that the inverse of an equivalence is an equivalence. b) Show that the (coordinatewise) composition of two “compatible” equivalences is an equivalence. 12. Let -À K Ä W\ be a transitive action and let B − \ . Show that - is equivalent to the action of left-translation by K on KÎstabÐBÑ. 13. Suppose that -À K Ä W\ and .À L Ä W] are equivalent transitive actions, under the equivalence Ðαß 0 Ñ. Prove that stabÐBÑ ¸ LC for any B − \ , where C œ 0 B.

Conjugacy 14. Let K be a group and let 1 − K. Show that Ø1K Ù is a normal subgroup of K. 15. Let K be a finite group and let 1 − K . Show that kGK Ð1Ñk   kKÎK w k where Kw is the commutator subgroup of K. 16. Let K be a finite group and let L Ÿ K with ÒK À LÓ œ #. Suppose that GK Ð2Ñ Ÿ L for all 2 − L . Prove that K Ï L is a conjugacy class of K. 17. Let K be a :-group and let L Ÿ K be a nonnormal subgroup of K and let + − K. Show that the number of conjugates of L that are fixed by every element of L + is positive and divisible by :. 18. a) Let K be a finite group and let L ü K. Show that 5ÐKÎLÑ œ 5ÐKÑ  5K ÐLÑ  " where 5K ÐLÑ is the number of K-conjugacy classes of L .

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231

b) Let K be a finite nonabelian group such that KÎ^ÐKÑ is abelian. Show that 5ÐKÑ   kKÎ^ÐKÑk  k^ÐKÑk  "

19. a)

Find all finite groups (up to isomorphism) that have exactly one conjugacy class. b) Find all finite groups (up to isomorphism) that have exactly two conjugacy classes. c) Find all finite groups (up to isomorphism) that have exactly three conjugacy classes. 20. a) If ; −  and 8  !, show that there are only finitely many solutions 5" ß á ß 58 in positive integers to the equation ;œ

" " â 5" 58

Hint: Use induction on 8. Look at the smallest denominator first. b) Show that for any integer 8  !, there are only finitely many finite groups (up to isomorphism) that have exactly 8 conjugacy classes. Hint: Use the class equation. 21. a) Let L be a proper subgroup of a finite group K. Show that the set W œ . L1 1−K

is a proper subset of K. b) If L is a proper subgroup of a group K and ÐK À LÑ  ∞, then the set W œ . L1 1−K

is a proper subset of K. 22. Let \ be a conjugacy class of K and let \ " œ ÖB" ± B − \×. a) Show that \ " is also a conjugacy class of K. b) Show that if K has odd order, then \ œ Ö"× is the only conjugacy class for which \ œ \ " . c) Show that if K has even order, then there is a conjugacy class \ other than Ö"× for which \ œ \ " . d) Show that if K is finite and 5ÐKÑ is even, then 9ÐKÑ is even. Show by example that the converse does not hold. 23. Let K be a group of order #7. Suppose that K has a conjugacy class of size 7. Prove that 7 is odd, and that K has an abelian normal subgroup of size 7. 24. Let L be normal in K and suppose that ÐK À LÑ œ : is a prime. Let B − L have the property that there is a 1 − K Ï L such that 1B œ B1. a) Show that kGK ÐBÑk œ :kGL ÐBÑk. b) Show that BL œ BK .

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Fundamentals of Group Theory

:-Groups and : -Subgroups 25. a)

26. 27. 28. 29. 30. 31.

32.

Let 0 À K Ä L be a group homomorphism. If K is a :-group, under what conditions, if any, is L a :-group? b) Let 0 À K Ä L be a group homomorphism. If L is a :-group, under what conditions, if any, is L a :-group? c) Let L ü K. If L and KÎL are both :-groups, under what conditions, if any, is K a :-group? Let K be a finite :-group. a) Prove that any cover of L  K has index :. b) Prove that a cover of the center ^ÐKÑ is abelian. Let L Ÿ K. Prove that K is a :-group if and only if L and KÎL are :groups. Let K be a finite simple nonabelian group. Show that 9ÐKÑ is divisible by at least two distinct prime numbers. Prove that the derived group Kw of a :-group K is a proper subgroup of K. Let K be a :-group. Show that if L Ÿ K and ÐK À LÑ  ∞, then ÐK À LÑ is a power of :. Let K œ  K: be a direct product of :-subgroups for distinct primes :. Show that if L Ÿ K, then L œ  ÐL ∩ K: Ñ. What if the primes are not distinct? Show that the generalized quaternion group U7 œ Ø+ß ,Ùß 9Ð+Ñ œ #7" ß 9Ð,Ñ œ %ß , # œ +#

7#

ß ,+, " œ +"

has only is single involution. 33. Prove that a finite :-group has the normalizer condition using the action of RK ÐLÑ on the conjugates conjK ÐLÑ of L by conjugation. 34. Let K be a nonabelian group of order :$ , where : is a prime. Determine the number 5ÐKÑ of conjugacy classes of K.

Additional Problems 35. Let : be a prime. a) Show that

kKPÐ8ß ™: Ñk œ Ð:8  "ÑÐ: 8  :ÑâÐ: 8  : 8" Ñ

b) Show that Y X Ð8ß ™: Ñ is a :-group, in fact,

kY X Ð8ß ™: Ñk œ :Ð8 8ÑÎ# #

c) Show that Y X Ð8ß ™: Ñ is a Sylow :-subgroup of KPÐ8ß ™: Ñ. d) For 8 œ # or 8  " Ÿ :, show that Y X Ð8ß ™: Ñ has exponent :. e) Show that Y X Ð$ß ™# Ñ ¸ U. 36. Let J be a finite field of size ; and let Z be an 8-dimensional vector space over J . Show that the number of subspaces of Z of dimension 5 is

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233

8 Ð; 8  "ÑÐ; 8  ;ÑâÐ; 8  ; 5" Ñ Š ‹ ³ 5 5 ; Ð;  "ÑÐ; 5  ;ÑâÐ; 5  ; 5" Ñ The expressions Ð 85 Ñ; are called Gaussian coefficients. Hint: Show that the number of 5 -tuples of linearly independent vectors in Z is Ð; 8  "ÑÐ; 8  ;ÑâÐ; 8  ; 5" Ñ 37. Let Lß O Ÿ K. a) Show that the distinct double cosets E1F , where + − K , form a partition of K. b) Show that kE1F k œ ÐE À F 1 ∩ EÑ.

Chapter 8

Sylow Theory

In 1872, the Norwegian mathematician Peter Ludwig Mejdell Sylow [32] published a set of theorems which are now known as the Sylow theorems. These important theorems describe the nature of maximal : -subgroups of a finite group, which are now called Sylow :-subgroups. (For convenience, we will collect the Sylow theorems into a single theorem.)

Sylow Subgroups We begin with the definition of a Sylow subgroup. Definition Let K be a group and let : be a prime. A Sylow :-subgroup of K is a maximal :-subgroup of K (under set inclusion). The set of all Sylow :subgroups of K is denoted by Syl: ÐKÑ. The number of Sylow :-subgroups of a group K is denoted by 8: ÐKÑ, or just 8: when the context is clear.… Of course, if a prime : divides 9ÐKÑ, then K contains a Sylow :-subgroup; in fact, every :-subgroup of K is contained in a Sylow :-subgroup. Also, if K is an infinite group and if L is a :-subgroup of K, then an appeal to Zorn's lemma shows that K has a Sylow :-subgroup containing L . Since conjugation is an order isomorphism and also preserves the group order of elements, it follows that if W is a Sylow : -subgroup of K, then so is every conjugate W + of W . Note also that if K is finite and 9ÐKÑ œ :8 7 where Ð:ß 7Ñ œ ", then any subgroup of order :8 is a Sylow :-subgroup. We will prove the converse of this a bit later: Any Sylow subgroup of K has order :8 .

The Normalizer of a Sylow Subgroup Let K be a finite group. If a Sylow :-subgroup W of K happens to be normal in K, then KÎW has no nonidentity :-elements. Hence, : y± ÐK À WÑ and so W is the

S. Roman, Fundamentals of Group Theory: An Advanced Approach, DOI 10.1007/978-0-8176-8301-6_8, © Springer Science+Business Media, LLC 2012

235

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Fundamentals of Group Theory

set of all :-elements of K . It also follows that W « K , since automorphisms preserve order. Of course, W is always normal in its normalizer RK ÐWÑ. Theorem 8.1 Let K be a finite group and let W − Syl: ÐKÑ. 1) W is the set of all :-elements of RK ÐWÑ. 2) Any :-element + − K Ï W moves W by conjugation, that is, W + Á W . 3) W is the only Sylow :-subgroup of RK ÐWÑ. 4) : y± ÐRK ÐWÑ À WÑ. 5) W « RK ÐWÑ.… If W − Syl: ÐKÑ, then W + Ÿ RK ÐWÑ+ for any + − K. Hence, if + normalizes RK ÐWÑ, then W + Ÿ RK ÐWÑ and since W + is also a Sylow :-subgroup of RK ÐWÑ, Theorem 8.1 implies that W + œ W . In other words, if + normalizes RK ÐWÑ, then + also normalizes W and so RK ÐRK ÐWÑÑ œ RK ÐWÑ Theorem 8.2 The normalizer RK ÐWÑ of a Sylow subgroup of K is selfnormalizing, that is, RK ÐRK ÐWÑÑ œ RK ÐWÑ

…

Soon we will be able to prove that not only is RK ÐWÑ self-normalizing, but so is any subgroup of K containing RK ÐWÑ.

The Sylow Theorems Let K be a finite group and let W − Syl: ÐKÑ. The fact that any : -element +  W moves W by conjugation prompts us to look at the action of a :-subgroup O of K by conjugation on the set conjK ÐWÑ œ ÖW + ± + − K× of conjugates of W in K. As to the stabilizer of W + , we have stabÐW + Ñ œ RK ÐW + Ñ ∩ O œ W + ∩ O and so

korbO ÐW + Ñk œ ÐO À W + ∩ OÑ

which is divisible by : unless O Ÿ W + , in which case the orbit has size ".

Sylow Theory

237

Hence, FixconjK ÐWÑ ÐOÑ œ ÖW + ± O Ÿ W + × and so

kconjK ÐWÑk ´ kÖW + ± O Ÿ W + ×k mod :

Now if O is a Sylow :-subgroup of K, then O Ÿ W + if and only if O œ W + and so kconjK ÐWÑk ´ œ

" mod : ! mod :

if O − conjK ÐWÑ if O Â conjK ÐWÑ

It follows that O  conjK ÐWÑ is impossible and so Syl: ÐKÑ œ conjK ÐWÑ is a conjugacy class and 8: ´ " mod : Note also that

8: œ kconjK ÐWÑk œ ÐK À RK ÐWÑÑ ± 9ÐKÑ

Finally, we can determine the order of a Sylow : -subgroup W , since ÐK À WÑ œ ÐK À RK ÐWÑÑÐRK ÐWÑ À WÑ and neither of the factors on the right is divisible by :. Hence, the order of W is the largest power of : dividing 9ÐKÑ. We have proved the Sylow theorems. Theorem 8.3 (The Sylow theorems [32], 1872) Let K be a finite group and let 9ÐKÑ œ :8 7, where : is a prime and : y± 7. 1) The Sylow :-subgroups of K are the subgroups of K of order :8 . 2) Syl: ÐKÑ is a conjugacy class in subÐKÑ. 3) The number 8: of Sylow :-subgroups satisfies 8: ´ " mod :

and

8: œ ÐK À RK ÐWÑÑ ± 9ÐKÑ

where W − Syl: ÐKÑ. 4) Let W − Syl: ÐKÑ. a) W is normal if and only if 8: œ ". b) W is self-normalizing if and only if 8: œ ÐK À WÑ œ 7, in which case all Sylow :-subgroups of K are self-normalizing. 5) If O is a :-subgroup of K, then kÖW − Syl: ÐKÑ ± O Ÿ W×k ´ " mod :

…

We will prove later in the chapter that every normal Sylow : -subgroup of a finite group is complemented. This is implied by the famous Schur–Zassenhaus theorem. However, we also have the following simple consequence of Theorem 3.1 concerning supplements of Sylow subgroups.

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Fundamentals of Group Theory

Theorem 8.4 Let K be a finite group. Then any Sylow : -subgroup of K and any subgroup whose index is a power of : are supplements.…

Sylow Subgroups of Subgroups Let K be a finite group and let L Ÿ K. We wish to explore the relationship between Syl: ÐKÑ and Syl: ÐLÑ. On the one hand, every W − Syl: ÐLÑ is contained in a X − Syl: ÐKÑ and so the set Syl: ÐWà KÑ œ ÖX − Syl: ÐKÑ ± W Ÿ X × is nonempty. Moreover, since E − Syl: ÐWà KÑ implies that E ∩ L is a :subgroup of L containing W , we have E − Syl: ÐWà KÑ

Ê

E∩L œW

In particular, the families Syl: ÐWà KÑ are disjoint, that is, W Á X − Syl: ÐLÑ

Ê

Syl: ÐWà KÑ ∩ Syl: ÐX à KÑ œ g

and so 8: ÐLÑ Ÿ 8: ÐKÑ On the other hand, if W − Syl: ÐKÑ, then the intersection W ∩ L need not be a Sylow :-subgroup of L , as can be seen by taking L and W to be distinct Sylow :-subgroups of K. However, if LW is a subgroup of K, then 9ÐWÑ ± 9ÐLWÑ and so kL ∩ W k

kLW k œ kL k kW k

where kLW kÎkW k is not divisible by :. Hence, kL ∩ W k and kL k are divisible by the same powers of : and so L ∩ W − Syl: ÐLÑ. Theorem 8.5 Let K be a finite group and let L Ÿ K . 1) If W − Syl: ÐLÑ, then E − Syl: ÐWà KÑ

Ê

E∩L œW

and W Á X − Syl: ÐLÑ

Ê

Syl: ÐWà KÑ ∩ Syl: ÐX à KÑ œ g

and so 8: ÐLÑ Ÿ 8: ÐKÑ

Sylow Theory

239

2) If W − Syl: ÐKÑ and LW Ÿ K, then W ∩ L − Syl: ÐLÑ

…

Some Consequences of the Sylow Theorems Let us consider some of the more-or-less direct consequences of the Sylow theorems.

A Partial Converse of Lagrange's Theorem A Sylow :-subgroup W of a group K has subgroups of all orders dividing 9ÐWÑ. This gives a partial converse to Lagrange's theorem. Theorem 8.6 Let K be a finite group and let : be a prime. If :5 ± 9ÐKÑ, then K has a subgroup of order :5 .…

More on the Normalizer of a Sylow Subgroup Recall that the normalizer RK ÐWÑ of a Sylow subgroup W of K is selfnormalizing. Now we can say more. Theorem 8.7 Let K be a finite group and let W − Syl: ÐKÑ. If W Ÿ RK ÐWÑ Ÿ L Ÿ K then L is self-normalizing. In particular, if L  K , then L is not normal in K . Proof. Conjugating by any + − RK ÐLÑ gives W + Ÿ RK ÐWÑ+ Ÿ L Ÿ K and so both W and W + are Sylow :-subgroups of L . It follows that W and W + are conjugate in L . Hence, there is an 2 − L for which W 2+ œ W , that is, 2+ − RK ÐWÑ Ÿ L . Thus, + − L and so RK ÐLÑ œ L .… The normalizer of a Sylow :-subgroup has a somewhat stronger property than is expressed in Theorem 8.7. In the exercises, we ask the reader to prove that RK ÐWÑ is abnormal.

Counting Subgroups in a Finite Group In an earlier chapter, we proved that if K is a :-group and :5 ± 9ÐKÑ, then the number 8:ß5 ÐKÑ of subgroups of K of order :5 satisfies 8:ß5 ÐKÑ ´ " mod : We have just proved that for any finite group K for which : ± 9ÐKÑ, 8: ÐKÑ ´ " mod : To see that (8.8) holds for all finite groups, we count the size of the set

(8.8)

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Y5 œ ÖÐLß WÑ ± L Ÿ Wß W − Syl: ÐKÑß 9ÐLÑ œ :5 × modulo :. On the one hand, for each W − Syl: ÐKÑ, there are 8:ß5 ÐWÑ ´ " subgroups of W of order :5 and so kY5 k ´ 8: ÐKÑ † " ´ "

On the other hand, for each L Ÿ K of order :5 , Theorem 8.3 implies that kÖW − Syl: ÐKÑ ± L Ÿ W×k ´ "

and so

kY5 k ´ 8:ß5 ÐKÑ † " ´ 8:ß5 ÐKÑ

Hence, 8:ß5 ÐKÑ ´ " and we have proved an important theorem of Frobenius. Theorem 8.9 (Frobenius [13], 1895) Let K be a group with 9ÐKÑ œ :8 7 where Ð7ß :Ñ œ ". Then for each " Ÿ 5 Ÿ 8, the number 8:ß5 ÐKÑ of subgroups of K of order :5 satisfies …

8:ß5 ÐKÑ ´ " mod :

When All Sylow Subgroups Are Normal Several good things happen when all of the Sylow subgroups of a group K are normal. In particular, let K be a finite group. In an earlier chapter (see Theorem 4.22 and Theorem 4.35), we showed that among the conditions: 1) 2) 3) 4)

Every subgroup of K is subnormal K has the normalizer condition Every maximal subgroup of K is normal KÎFÐKÑ is abelian

the following implications hold: 1) Í 2) Ê 3) Í 4) We also promised to show that these four conditions are equivalent, which we can do now, adding several additional equivalent conditions into the bargin. First, let us speak about arbitrary (possibly infinite) groups. If K is a group, let Ktor denote the set of all torsion (periodic) elements of K. If K is abelian, then Ktor is a subgroup of K. However, in the nonabelian general linear group KPÐ#ß ‚Ñ, the elements EœŒ

! "

" !

and F œ Œ

! "

" "

Sylow Theory

241

are torsion but their product is not. Hence, Ktor is not always a subgroup of K. Note that when Ktor is a subgroup of K, then Ktor « K since automorphisms preserve order. Theorem 8.10 Let K be a group in which every Sylow subgroup is normal. Let the Sylow subgroups of K be Ö]: ± : − c ×. Then Ktor œ  ]: :−c

and so Ktor Ÿ K. Thus, the product of two elements of finite order has finite order. Proof. Since the Sylow :-subgroups are normal and pairwise essentially disjoint, they commute elementwise. In particular, if +" ß á ß +8 come from distinct Sylow subgroups, then 9Ð+" â+8 Ñ œ 9Ð+" Ñâ9Ð+8 Ñ and so the family of Sylow subgroups is strongly disjoint and ] ³  ]: © Ktor :−c

/7 For the reverse inclusion, if + − Ktor has order 8 œ :"/" â:7 where the primes :3 are distinct, then Corollary 2.11 implies that + œ +" â+7 , where 9Ð+5 Ñ œ :5/5 and so +5 − ]:5 , whence + − ] .…

Now we turn to finite groups in which all Sylow subgroups are normal. Theorem 8.11 Let K be a finite group, with Sylow subgroups Ö]: ± : − c × . The following are equivalent: 1) Every Sylow subgroup of K is normal. 2) K is the direct product of its Sylow :-subgroups K œ  ]: :−c

3) If L Ÿ K, then L œ  ÐL ∩ ]: Ñ :−c

4) K is the direct product of :-subgroups. 5) (Strong converse of Lagrange's theorem) If 8 ± 9ÐKÑ, then K has a normal subgroup of order 8. 6) Every subgroup of K is subnormal. 7) K has the normalizer condition. 8) Every maximal subgroup of K is normal. 9) KÎFÐKÑ is abelian. If these conditions hold, then K has the center-intersection property. In particular, ^ÐKÑ is nontrivial.

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Proof. Theorem 8.10 shows that 1) implies 2) and the converse is clear. If 1) holds, then L]: Ÿ K and so the Sylow :-subgroups of L are ÖL ∩ ]: ± : − c ×. Moreover, since L ∩ ]: ü L , it follows that L is the direct product of its Sylow :-subgroups and so 3) holds. It is clear that 3) implies 2) and so 1)–3) are equivalent. Also, it is clear that 2) Ê 4). If 4) holds and : is a prime dividing 9ÐKÑ, then we can isolate the factors in the direct product of K that have exponent :, say KœT U where T is a direct product of :-subgroups and U is a direct product of ; subgroups for various primes ; Á :. Then T is the set of all :-elements of K and so T a Sylow :-subgroup of K. But T ü K and so 1) holds. Thus, 1)–4) are equivalent. It is clear that 5) Ê 1). To see that 2) Ê 5), any divisor 8 of 9ÐKÑ is a product 8 œ #.: where .: ± 9Ð]: Ñ and since ]: has a normal subgroup of order .: , the direct product of these subgroups is a normal subgroup of K of order 8. Thus, 1)–5) are equivalent and we have already remarked that 6) Í 7) Ê 8) Í 9) To see that 3) Ê 6), if L  K, then one of the factors L ∩ ]: is proper in ]: and so there is a subgroup R: for which L ∩ ]: – R: Ÿ ] : Hence, L –  ÐL ∩ ]; Ñ  R: ;Á:

Since L is an arbitrary proper subgroup of K, it follows that all subgroup of K are subnormal. To see that 6) Ê 1), if ]: is not normal in K, then since RK Ð]: Ñ  K is subnormal, there is a subgroup L: for which RK Ð]: Ñ – L: Ÿ K. But this contradicts the fact that RK Ð]: Ñ is self-normalizing. Hence, ]: – K. Thus, 1)–7) are equivalent and imply 8). Similarly, if 8) holds but ]: is not normal in K, then there is a maximal subgroup Q Ÿ K for which ]: Ÿ RK Ð]: Ñ Ÿ Q – K which contradicts Theorem 8.7. Hence, ]: ü K and 1) holds. Finally, if 3) holds and R Ÿ K is nontrivial, then R ∩ ]; ü ]; is nontrivial for some ; − c and R ∩ ^ÐKÑ œ  ÐR ∩ ^ÐKÑ ∩ ]: Ñ :−c

Sylow Theory But R ∩ ^ÐKÑ ∩ ]; œ R ∩ ^Ð]; Ñ R ∩ ^ÐKÑ.…

is

nontrivial

and

therefore

243 so

is

The hypotheses of the previous theorems hold for all abelian groups. Corollary 8.12 Let K be an abelian group. 1) (Primary decomposition) Then Ktor is the direct product of its Sylow :subgroups. 2) (Converse of Lagrange's theorem) If K is finite and 8 ± 9ÐKÑ, then K has a subgroup of order 8.… We will add one additional characterization (nilpotence) to Theorem 8.11 in a later chapter (see Theorem 11.8).

When a Subgroup Acts Transitively; The Frattini Argument The Frattini argument (Theorem 7.2) shows that if a group K acts on a nonempty set \ and if L Ÿ K is transitive on \ , then K œ L stabK ÐBÑ and if L is regular on \ , then K œ L ì stabK ÐBÑ and so stabK ÐBÑ is a complement of L in K. To apply this idea, let K be a finite group and let WŸLüK where W − Syl: ÐLÑ. Let K act on conjK ÐWÑ by conjugation. Since W + − Syl: ÐLÑ for any + − K, it follows that L acts transitively on conjK ÐWÑ. Hence, K œ L stabK ÐWÑ œ LRK ÐWÑ This specific argument is also referred to as the Frattini argument. Theorem 8.13 Let K be a finite group and let L Ÿ K . If W − Syl: ÐLÑ , then K œ LRK ÐWÑ and if the action of L by conjugation on conjK ÐWÑ is regular, then K œ L ì RK ÐWÑ

…

This theorem can be used to show that the Frattini subgroup of a finite group K has the property that all of its Sylow subgroups are normal in K.

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Fundamentals of Group Theory

Theorem 8.14 (Frattini [12], 1885) If K is a finite group, then the Frattini subgroup FÐKÑ has the property that all of its Sylow subgroups are normal in K. Proof. If W − Syl: ÐFÑ, then W Ÿ F ü K and the Frattini argument shows that K œ FRK ÐWÑ But if RK ÐWÑ  K, then there is a maximal subgroup Q of K for which RK ÐWÑ Ÿ Q and so K Ÿ Q , a contradiction. Hence, RK ÐWÑ œ K and W ü K.…

The Search for Simplicity The Sylow theorems, along with group actions and counting arguments, provide powerful tools for the analysis of finite groups. A key issue with respect to finite groups is the question of simplicity. As we will discuss in a later chapter, the issue of which finite groups (up to isomorphism) are simple appears to be resolved, but the resolution is so complex that some mathematicians may still have questions regarding its completeness and its accuracy. We have seen that a group of prime-power order :8 has a normal subgroup of each order :5 ± :8 . Accordingly, we will do no further direct analysis of :groups in this chapter. Throughout our discussion, : will denote a prime, ]: will denote an arbitrary Sylow :-subgroup and, as always, 8: denotes the number of Sylow :-subgroups of K. Recall that 1) 8: œ "  5: for some integer 5   !. 2) 8: œ ÐK À RK Ð]: ÑÑ ± 9ÐKÑ. Note that if 9ÐKÑ œ :8 7, where : y± 7, then 8: ± 9ÐKÑ if and only if 8: ± 7. The following facts (among others) are useful in showing that a group is not simple: 3) 4) 5) 6)

]: is normal if and only if 8: œ ". If ÐK À LÑ is equal to the smallest prime dividing 9ÐKÑ, then L – K. The kernel of any action -À K Ä W\ is a normal subgroup of K. If L  K, then ÐK À L ‰ Ñ ± ÐK À LÑx. Hence, if 9ÐKÑ y± ÐK À LÑx, then L ‰ is a nontrivial proper normal subgroup of K.

We will also have use for the fact that if : is prime and " Ÿ /  : , then /: is the smallest integer for which :/ ± Ð/:Ñx.

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245

The 8: -Argument It happens quite often that for some odd prime : ± 9ÐKÑ, the integers "  5: do not divide 9ÐKÑ unless 5 œ !, in which case 8: œ " and ]: ü K . Let us refer to the argument 8: œ "  5: ± 9ÐKÑ

Ê

5œ!

as the 8: -argument. Note that the 8: -argument does not hold if : œ #, unless 9ÐKÑ is a power of #, since "  #5 ± 9ÐKÑ for some 5  ". Example 8.15 If 9ÐKÑ œ **)# œ # † ( † #$ † $", then routine calculation shows that the 8( argument holds: "  (5 ± 9ÐKÑ

Ê

5œ!

and so ]( – K and K is not simple.… Example 8.16 If 9ÐKÑ œ :8 7 for 8   ", 7  " and : y± 7, then 8: œ Ð"  5:Ñ ± 7 and so if 7  :, then 5 œ !, whence ]: – K . Thus, groups of order :8 ß #:8 ß á ß Ð:  "Ñ:8 for : prime and 8   " have ]: – K and so are not simple.… A little programming shows that among the orders up to "!!!! (not including prime powers) there are only &'* orders (less than '%) that are not succeptible to the 8: -argument for some :. Thus, the vast majority of orders up to "!!!! are either prime powers or have the property that groups of that order have a normal Sylow :-subgroup.

Counting Elements of Prime Order If : is a prime and : ± 9ÐKÑ but :# y± 9ÐKÑ, then each of the 8: distinct Sylow : subgroups of K has order : and so the subgroups are pairwise essentially disjoint. Hence, K contains exactly 8: † Ð:  "Ñ distinct elements of order : . Sometimes this simple counting of elements (for different primes :) is enough to show than one of the Sylow subgroups is normal. Example 8.17 Let 9ÐKÑ œ $! œ # † $ † &. Then based on the fact that 8: œ "  5: ± 9ÐKÑ, we can conclude only that 8$ − Ö"ß "!× and 8& − Ö"ß '×. However, if 8$ œ "! and 8& œ ', then K contains at least 8$ † Ð$  "Ñ œ #! elements of order $ and #% elements of order &, totalling %% elements. Hence, one of ]$ or ]& must be normal in K.…

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Index Equal to Smallest Prime Divisor If 9ÐKÑ œ :; 5 where :  ; are primes, then ]; – K, because ÐK À ]; Ñ œ : is the smallest prime dividing 9ÐKÑ. Moreover, it is clear that K œ ]; z ]: Example 8.18 If 9ÐKÑ œ $ † &# œ (&, then ]& – K and K œ ]$ z ]& Also, "  $5 ± #& holds only for 5 œ ! or 5 œ ) and so 8$ œ " or 8$ œ #&. Note that if 8$ œ ", then K œ ]$  ]& is abelian.… When 9ÐKÑ œ :; , we can give a fairly complete analysis as follows. Theorem 8.19 Let 9ÐKÑ œ :; , with :  ; primes. Then K œ G; Ð,Ñ z G: Ð+Ñ where ,+ œ ,5 for some " Ÿ 5  ; and 5 : ´ " mod ;. Moreover, K is cyclic if and only if : y± ;  ". Proof. We have seen that K œ ]; z ]: œ G; Ð,Ñ z G: Ð+Ñ Thus, +,+" œ , 5 for some " Ÿ 5  ; and repeated conjugation by + gives , œ +: ,+: œ , 5

:

which implies that 5 : ´ " mod ; . Moreover, 8: ± ; and so 8: œ " or 8: œ ; . But 8: œ " if and only if ]: ü K, that is, if and only if K is cyclic and 8: œ ; if and only if "  5: œ ; , that is, if and only if : y± ;  ".… Example 8.20 Let us return to the case 9ÐKÑ œ $! œ # † $ † &. We saw in Example 8.17 that one of ]$ or ]& must be normal in K. It follows that ]$ ]& ü K has order "& and so is cyclic. Hence, ]$ ß ]& « ]$ ]& ü K and so both ]$ and ]& are normal in K.…

Using the Kernel of an Action The kernel of an action -À K Ä W\ is normal in K and this can be a useful technique for finding normal subgroups, although they need not be Sylow subgroups. For example, if K acts on Syl: ÐKÑ by conjugation, then the representation map -À K Ä W5:" has kernel

Sylow Theory ,

Oœ

247

RK Ð] Ñ

] −Syl: ÐKÑ

which is a normal subgroup of K. The problem is that it may be either trivial or equal to K. Let 9ÐKÑ œ :7 ? and 9ÐOÑ œ := @, where 7   "ß ?  " and : y± ?. Also, let 8: œ 5:  ". It is clear that O œ K is equivalent to 8: œ " and implies that = œ 7. Conversely, if = œ 7, then O contains a Sylow : -subgroup W of K. But the only Sylow :-subgroup of K in RK Ð] Ñ is ] itself and so 8: œ " and O œ K. Thus, OœK

Í

]: – K

Í

=œ7

As to the nontriviality of O , the induced embedding of KÎO into W5:" implies that :7=

? ¹ Ð5:  "Ñx @

and so :7= ± Ð5:Ñx. Hence, if 5  :, then 7  5 Ÿ =. It follows that if 5  7, then =  ! and O is nontrivial. Thus, 5  minÖ7ß :×

Ê

O Á Ö"×

We note finally that O has a somewhat simpler form if 8: œ ?, since then each ]: is self-normalizing and ,

Oœ

]

] −Syl: ÐKÑ

Theorem 8.21 Let 9ÐKÑ œ :7 ? where : is prime, 7   ", ?  " and : y± ?. Let 8: œ "  5:. 1) If 5 œ !, then ]: – K. 2) If !  5  minÖ7ß :×, then Oœ

,

RK Ð] Ñ

] −Syl: ÐKÑ

is a nontrivial proper normal subgroup of K of order := @, where 7  5 Ÿ = Ÿ 7  " and @ ± ?. In addition, if 5 œ Ð?  "ÑÎ:, then Oœ

,

]

] −Syl: ÐKÑ

has order := .… Example 8.22 If 9ÐKÑ œ "!) œ $$ † %, then "  $5 ± % and so 5 œ ! or 5 œ ". Thus, this case is not amenable to the 8: -argument. However, if 5 œ " then

248

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Theorem 8.21 implies that

,

Oœ

]

] −Syl$ ÐKÑ

is a nontrivial proper normal subgroup of K of order *. Thus, K is not simple. If 9ÐKÑ œ ")* œ $$ † (, then "  $5 ± ( and so 5 œ ! or 5 œ #. If 5 œ #, then Theorem 8.21 implies that ,

Oœ

]

] −Syl$ ÐKÑ

is a nontrivial proper normal subgroup of K of order $ or *. If 9ÐKÑ œ $!! œ ## † $ † &# , then 8& œ "  &5 ± "# and so 5 œ ! or 5 œ " (and 8& œ '). But if 5 œ ", then Theorem 8.21 implies that K is not simple.… Even when O is trivial and the previous theorem does not apply, we learn that K ä W5:" , which can sometimes be useful. Example 8.23 If 9ÐKÑ œ :Ð:  "Ñ, where : is prime. Then 8: œ " or 8: œ :  ". While the previous theorem does not apply, if 8: œ :  ", then Oœ

,

] œ Ö"×

] −Syl: ÐKÑ

Hence, K ä W:" . As an example, if 9ÐKÑ œ "# œ $ † %, then either ]$ – K or y E% K ä W% . But 9ÐKÑ œ 9ÐE% Ñ and so in the latter case, K ¸ E% . Thus, if K ¸ then ]$ – K. We will use this fact later to determine all groups of order "#.…

The Normal Interior If L  K, we have seen that ÐK À L ‰ Ñ ± ÐK À LÑx and so if 9ÐKÑ y± ÐK À LÑx, then L ‰ is a nontrivial proper normal subgroup of K . Hence, if /7 9ÐKÑ œ :"/" â:7

where :"  â  :7 are primes and 7   #, then for any 5 , /5  :5 ß ÐK À LÑ  /5 :5

Ê

:5/5 y± ÐK À LÑx

Ê

9ÐKÑ y± ÐK À LÑx

and so L ‰ – K is nontrivial. /7 Theorem 8.24 Let 9ÐKÑ œ :"/" â:7 where :"  â  :7 are primes and 7   #. Suppose that /5  :5 for some " Ÿ 5 Ÿ 7.

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249

1) If L  K has index ÐK À LÑ  /5 :5 , then L ‰ is a nontrivial proper normal subgroup of K. 2) In particular, if "  8:3  /5 :5 , then RK Ð]:3 щ is a nontrivial proper normal subgroup of K.… Example 8.25 Let 9ÐKÑ œ '#!" œ $# † "$ † &$. Then 8$ œ "  $5 ± "$ † &$, which implies that 8$ − Ö"ß "$×. If 8$ œ ", then ]$ – K . If 8$ œ "$  &$, then Theorem 8.24 implies that RK Ð]$ щ is a nontrivial proper normal subgroup of K. Thus K is not simple.…

Using the Normalizer of a Sylow Subgroup Let : Á ; be primes dividing 9ÐKÑ and let ]; − Syl; ÐKÑ. Under the assumption that 8;  " and so RK Ð]; Ñ  K, suppose that : y± 8; , that is, : ± 9ÐRK Ð]; ÑÑ and that T − Syl: ÐRK Ð]; ÑÑ. There are various things we can say about 9ÐRK ÐT ÑÑ. First, if T – RK Ð]; Ñ, then RK Ð]; Ñ Ÿ RK ÐT Ñ and so 9ÐRK Ð]; ÑÑ ± 9ÐRK ÐT ÑÑ On the other hand, even if T is not normal in RK Ð]; Ñ, the fact that ]; ü RK Ð]; Ñ implies that T ]; is a subgroup of K. Hence, if T ]; is abelian, then ]; Ÿ RK ÐT Ñ and so 9Ð]; Ñ ± 9ÐRK ÐT ÑÑ In either case, if T is not a Sylow : -subgroup of K but T – T ‡ − Syl: ÐKÑ, then T ‡ Ÿ RK ÐT Ñ, whence 9ÐT ‡ Ñ ± 9ÐRK ÐT ÑÑ These conditions tend to make RK ÐT Ñ large. Example 8.26 If 9ÐKÑ œ $'(& œ $ † &# † (# , then it is easy to see that 8( − Ö"ß "&×. If 8( œ "&, then 9ÐRK Ð]( ÑÑ œ & † (# . Let T be a Sylow &subgroup of RK Ð]( Ñ. The number of such subgroups is "  &5 ± (# and so T – RK Ð]( Ñ. Hence, & † (# ± 9ÐRK ÐT ÑÑ Also, T has index & in T ‡ − Syl& ÐKÑ and so T – T ‡ , whence &# ± 9ÐRK ÐT ÑÑ and so &# † (# ± 9ÐRK ÐT ÑÑ Hence, either RK ÐT Ñ œ K , in which case T – K or else RK ÐT Ñ has index $ in K and so is normal in K.…

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Fundamentals of Group Theory

Suppose that 9ÐKÑ œ :;?, where : Á ; are primes that do not divide ?. If : y± 8; , that is, if : ± 9ÐR Ð]; ÑÑ, then ]: Ÿ R Ð]; Ñ and so ]: ]; Ÿ K has order :; . Hence, if : y± Ð;  "Ñ, then ]: ]; is abelian (cyclic) and so ]; Ÿ R Ð]: Ñ. Thus, ; ± 9ÐR Ð]: ÑÑ and so 8: ± 9ÐKÑÎ:; . Theorem 8.27 If 9ÐKÑ œ :;?, where :  ; are primes that do not divide ?, then : y± Ð;  "Ñ and

: y± 8;

Ê

8: ¹

9ÐKÑ :;

…

Example 8.28 If 9ÐKÑ œ "()& œ $ † & † ( † "(, then a routine calculation gives 8$ − Ö"ß (ß )&ß &*&×

and 8"( − Ö"ß $&×

But $  "(ß

$ y± Ð"(  "Ñß

$ y± 8"(

Ê

8$ ¹

9ÐKÑ œ $& $ † "(

and so 8$ œ " or 8$ œ (. Hence, Theorem 8.24 now implies that one of ]$ or R Ð]$ щ is a proper nontrivial normal subgroup of K.…

Groups of Small Order We have already examined the groups of order %, ' and ). Let us now look at all groups of order "& or less. Of course, all groups of prime order are cyclic. We will again denote an arbitrary Sylow : -subgroup of K by ]: .

Groups of Order % The groups of order % are (up to isomorphism): 1) G% , the cyclic group 2) Z ¸ G# } G# , the Klein 4-group.

Groups of Order ' The groups of order ' are (up to isomorphism): 1) G' , the cyclic group 2) H' ¸ W$ , the nonabelian dihedral (and symmetric) group.

Groups of Order ) The groups of order ) are (up to isomorphism): 1) G) , the cyclic group 2) G% } G# , abelian but not cyclic 3) G# } G# } G# , abelian but not cyclic

Sylow Theory

251

4) H) , the (nonabelian) dihedral group 5) U, the (nonabelian) quaternion group.

Groups of Order * Theorem 7.9 implies that the groups of order * are (up to isomorphism): 1) G* , the cyclic group 2) G$ } G$ , abelian but not cyclic.

Groups of Order "! If 9ÐKÑ œ "! œ # † &, then Theorem 8.19 implies that K œ Ø+ß ,Ùß 9Ð+Ñ œ #ß 9Ð,Ñ œ &ß +,+" œ , 5 where 5 # ´ " mod &, that is, 5 œ " or %. In the former case, + and , commute and K is cyclic. In the latter case, K œ H"! . Thus, the groups of order "! are (up to isomorphism): 1) G"! ¸ G& } G# , the cyclic group 2) H"! , the nonabelian dihedral group.

Groups of Order "# We have accounted for three nonabelian groups of order "#: the alternating group E% , the dihedral group H"# and the semidirect product X œ G$ Ð+Ñ z G% Ð,Ñ, where ,+, " œ +# We wish to show that this completes the list of nonabelian groups of order "#. y E% . Then Example 8.23 shows that ]$ œ G$ Ð+Ñ ü K and so Assume that K ¸ K œ G$ Ð+Ñ z ]# , where ]# œ G% Ð,Ñ or ]# œ G# ÐBÑ  G# ÐCÑ. If K œ G$ Ð+Ñ z G% Ð,Ñ, then ,+, " œ +

or ,+, " œ +#

In the former case K is abelian and in the latter case K œ X . If K œ G$ Ð+Ñ z ÐG# ÐBÑ  G# ÐCÑÑ then +B œ + or +#

and +C œ + or +#

We consider three cases. If +B œ + and +C œ +, then K is abelian. If +B œ + and +C œ +# , then 9Ð+BÑ œ 9Ð+Ñ9ÐBÑ œ ' and Ð+BCÑ# œ +BC+BC œ +CÐB+BÑC œ +C+C œ +$ œ "

252

Fundamentals of Group Theory

Hence, Ø+BCß CÙ is generated by two distinct involutions whose product has order ' and so Theorem 2.36 implies that K œ H"# . Finally, if +B œ +# œ +C , then Ð+BÑ# œ +B+B œ +$ œ " and +BC œ Ð+# ÑC œ +% œ + and so 9Ð+BCÑ œ 9Ð+Ñ9ÐBCÑ œ '. Thus, Ø+Bß CÙ is dihedral and K œ H"# . Thus, the groups of order "# are (up to isomorphism) 1) 2) 3) 4) 5)

G"# ¸ G% } G$ , the cyclic group G# } G# } G$ , abelian but not cyclic E% , the nonabelian alternating group H"# , the nonabelian dihedral group X , the nonabelian group described above.

Groups of Order "% If 9ÐKÑ œ "% œ # † (, then Theorem 8.19 implies that K œ Ø+ß ,Ùß 9Ð+Ñ œ #, 9Ð,Ñ œ (ß +,+" œ , 5 where 5 # ´ " mod (, that is, 5 œ " or '. In the former case, K is cyclic. In the latter case, K is dihedral. Thus, the groups of order "% are (up to isomorphism): 1) G"% ¸ G( } G# , the cyclic group 2) H"% , the nonabelian dihedral group.

Groups of Order "& Theorem 8.19 implies that all groups of order "& are cyclic.

On the Existence of Complements: The Schur–Zassenhaus Theorem In this section, we use group actions to prove the Schur–Zassenhaus Theorem, which gives a simple sufficient (but not necessary) condition under which a normal subgroup L of a group K has a complement O , thus giving a semidirect decomposition K œ L z O . Definition Let K be a finite group. A Hall subgroup L of K is a subgroup with the property that its order 9ÐLÑ and index ÐK À LÑ are relatively prime.… The Schur–Zassenhaus Theorem states that a normal Hall subgroup L has a complement. The tool that we will use in proving the Schur–Zassenhaus Theorem is the Frattini argument (Theorem 7.2). In particular, we consider the action of left translation by K on the set e of all right transversals of L and show that this action is regular, whence K œ L z stabK ÐVÑ for any V − e. So let us take a closer look at transversals and their actions.

Sylow Theory

253

Transversals and Their Actions Let K be a finite group and let L be a normal Hall subgroup of K, with right cosets LÏK œ ÖL œ L" ß á ß L7 × Let e be the set of all right transversals of L . If V œ Ö<" ß á ß <7 × − e where <3 − L3 for all 3 and if + − K, then L+<3 œ +L<3 and so the cosets L+<3 are distinct. Hence, +V œ Ö+<" ß á ß +<7 × − e and it is clear that K acts on e by left translation. Although L need not act transitively on e, we can raise the action of K to an action on the congruence classes of an appropriate K-congruence on e so that L does act transitively. The K -congruence condition is V´W

Ê

+V ´ +W

for all + − K, that is, Ö<" ß á ß <7 × ´ Ö=" ß á ß =7 ×

Ê

Ö+<" ß á ß +<7 × ´ Ö+=" ß á ß +=7 ×

This leads us to try the following. Assuming that V and W are indexed so that <3 and =3 are in the same right coset L3 , define a binary relation ´ by $<3 =" 3 œ" 7

V´W

if

3œ"

Letting VlW œ $<3 =" 3 7

3œ"

the definition becomes V´W

if

VlW œ "

This relation is clearly reflexive. Also, since <3 =" 3 − L for all 3, if L is abelian, then for any Vß Wß X − e, ÐVlWÑ" œ WlV

and ÐVlWÑÐWlX Ñ œ VlX

and so ´ is an equivalence relation on e. Let eÎ ´ denote the set of equivalence classes of e and let ÒVÓ denote the equivalence class containing V .

254

Fundamentals of Group Theory

Note that if 2 − L , then <3 =" 3 − L implies that 7 Ð2VÑlW œ $2<3 =" 3 œ 2 ÐVlWÑ 7

3œ"

Moreover, since <3 and =3 are in the same right coset of L , so are +<3 and +=3 and so for any + − K, +Vl+W œ $Ð+<3 ÑÐ+=3 Ñ" œ + $<3 =" +" œ ÐVlWÑ+ 3 7

7

3œ"

3œ"

Hence, +V ´ +W if and only if V ´ W . Thus, ´ is a K -congruence on e and the induced action is +ÒVÓ œ Ò+VÓ Now, L acts transitively if and only if for all Vß W − e, there is an 2 − L for which Ò2VÓ œ ÒWÓ, that is, for which " œ Ð2VÑlW œ 2 7 ÐVlWÑ or equivalently, 27 œ VlW But this equation always has a solution in L since 7 œ ÐK À LÑ and 9ÐLÑ are relatively prime. The action of L on eÎ ´ is also regular, since Ò2VÓ œ ÒVÓ if and only if " œ Ð2VÑlV œ 2 7 ÐVlVÑ œ 2 7 which implies that 2 œ ". Hence, if L is a normal abelian subgroup of K , the Frattini argument implies that K œ L z stabK ÐÒVÓÑ for any V − e. As to the conjugacy statement, any conjugate of a complement of L is also a complement of L . On the other hand, if K œL zO then Theorem 5.3 implies that O − e and so L z O œ K œ L z stabK ÐÒOÓÑ But O Ÿ stabK ÐÒOÓÑ and so O œ stabK ÐÒOÓÑ

Sylow Theory

255

Thus, the set of complements of L in K is precisely the set ÖstabK ÐÒVÓÑ ± V − e× Also, since K acts transitively on eÎ ´ , all stabilizers are conjugate and so all complements of L are conjugate. We have proved the abelian version of the Schur–Zassenhaus Theorem. Theorem 8.29 (Schur–Zassenhaus Theorem—abelian version) If K is a finite group, then any normal abelian Hall subgroup L of K has a complement in K. Moreover, the complements of L in K form a conjugacy class of subÐKÑ.…

The Schur–Zassenhaus Theorem The condition of abelianness can be removed from the Schur–Zassenhaus Theorem. The proof of the general Schur–Zassenhaus theorem uses the Frattini argument as well, but also uses the Feit–Thompson theorem to establish the conjugacy portion of the theorem. Let us isolate the portion that uses the Feit– Thompson Theorem. Theorem 8.30 Let K be a nontrivial group of odd order. 1) K w  K 2) K has a normal subgroup O of prime index. Proof. Part 1) can be proved by induction on 9ÐKÑ. If 9ÐKÑ œ $, then K is abelian and Kw œ Ö"×  K . Assume the result holds for groups of odd order less than 9ÐKÑ and let 9ÐKÑ be odd. Then the Feit–Thompson theorem implies that K is either abelian or nonsimple. If K is abelian, then Kw œ Ö"×  K . If K is nonsimple, then there is a Ö"× – O – K and so KÎO has odd order less than 9ÐKÑ. Hence, the inductive hypothesis implies that ÐKÎOÑw  KÎO , which implies that Kw  K . For part 2), since Kw  K, it follows that K has a maximal normal subgroup O containing Kw . Then KÎO is simple and abelian and so has prime order.… Theorem 8.31 (Schur–Zassenhaus theorem) Any normal Hall subgroup L of finite group K has a complement in K. Moreover, the complements of L in K form a conjugacy class in K. In particular, any normal Sylow subgroup of K is complemented. Proof. We may assume that L is nontrivial and proper. The proof of the existence of a complement is by induction on 9ÐKÑ. The result is true if 9ÐKÑ œ ". Assume it is true for all orders less than 9ÐKÑ. Let L ü Hall K denote the fact that L is a normal Hall subgroup of K. If L has a proper supplement O , that is, if K œ LO for some O  K, then M œ L ∩ O ü Hall O and so the inductive hypothesis implies that O œ M z N , whence

256

Fundamentals of Group Theory

K œ LO œ LÐM z N Ñ œ L z N and so L is complemented. But since L is a Hall subgroup, if : is a prime dividing 9ÐLÑ, then any W − Syl: ÐLÑ is also a Sylow : -subgroup of K and the Frattini argument implies that K œ LR where R œ RK ÐWÑ. Thus, if R  K, then L is complemented. On the other hand, if R œ K, then W is a normal Sylow :-subgroup of K and so W ö K, whence ^ ³ ^ÐWÑ ö K. Since LÎ^ ü Hall KÎ^ , the inductive hypothesis implies that K L O œ z ^ ^ ^ for some O Ÿ K and so O is a supplement of L . But if O  K, then L is complemented and if O œ K, then L œ ^ is abelian and so L is complemented in this case as well. We now turn to the statement about conjugacy. Since any conjugate of a complement of L is also a complement of L , we are left with showing that any two complements O" and O# of L are conjugate. The proof is by induction on 9ÐKÑ. Of course, the result is true if 9ÐKÑ œ ". Assume the result is true for groups of order less than 9ÐKÑ. If L contains a nontrivial proper subgroup R that is normal in K, then O3 R ÎR is a complement of LÎR in KÎR for each 3 and so by the inductive hypothesis, O# R O" R œŒ R R

+R

œ

O"+ R R

for some + − K. Hence, R z O# œ R z O"+ and so the inductive hypothesis applied to the group R z O#  K implies that O# and O"+ are conjugates, whence so are O# and O" . If L does not contain a nontrivial proper subgroup that is normal in K, then we can easily dispatch the case 9ÐLÑ odd with the help of Theorem 8.30, which implies that L w ö L – K and so L w – K , whence L w œ Ö"×, that is, L is abelian. Then the abelian version of the Schur–Zassenhaus Theorem completes the proof. So we may assume that 9ÐLÑ is even, which implies that 9ÐKÎLÑ is odd.

Sylow Theory

257

Then Theorem 8.30 implies that K has a normal subgroup E for which L Ÿ E – K and ÐK À EÑ œ : is prime. Hence, E œ L z ÐO" ∩ EÑ and

E œ L z ÐO# ∩ EÑ

and the inductive hypothesis implies that O# ∩ E œ ÐO" ∩ EÑ+ for some + − E. But ÐO3 À O3 ∩ EÑ œ ÐO3 E À EÑ œ ÐK À EÑ œ : and so O3 ∩ E is a supplement of any ]3 − Syl: ÐO3 Ñ, that is, O3 œ ÐO3 ∩ EÑ]3 Conjugating this for 3 œ " by + gives O"+ œ ÐO# ∩ EÑ]"+ and so it is clear that we need to relate ]"+ to ]# . But O" ∩ E ü O" implies that ]" Ÿ O" Ÿ RK ÐO" ∩ EÑ and so ]"+ Ÿ RK ÐO" ∩ EÑ+ œ RK ÐO# ∩ EÑ Hence, ]# and ]"+ are Sylow :-subgroups of RK ÐO# ∩ EÑ, whence ]",+ œ ]# for some , − RK ÐO# ∩ EÑ. Conjugating by , gives O",+ œ ÐO# ∩ EÑ]",+ œ ÐO# ∩ EÑ]# œ O# as desired.… The Schur–Zassenhaus Theorem leads to the following important corollary. Corollary 8.32 Let 9ÐKÑ œ 87 where Ð8ß 7Ñ œ ". If K has a normal (Hall) subgroup R of order 8, then any subgroup L of K that has order 7w dividing 7 is contained in some complement of R . Proof. The Schur–Zassenhaus Theorem implies that there is a O Ÿ K for which K œ R z O . Then kR L ∩ O k œ 7w and R z L œ R z ÐR L ∩ OÑ Hence, the Schur–Zassenhaus Theorem implies that there exists + − K for which L œ ÐR L ∩ OÑ+ Ÿ O + But O + is also a complement of R in K .…

258

Fundamentals of Group Theory

*Sylow Subgroups of W8 In this section, we determine the Sylow subgroups of the symmetric group W8 in terms of wreath products. First, we need to compute the order of a Sylow :subgroup of W8 . Theorem 8.33 Let : be a prime dividing 8x and let 8 œ +!  + " :  â  + 7 : 7 be the base-: representation of 8, that is, ! Ÿ +5  :. The largest exponent / of : for which :/ ± 8x is 7

PÐ7Ñ œ

+5 Q Ð5Ñ 5œ"

where Q Ð5Ñ œ :5  :5"  â  " œ

:5  " :"

and so the order of a Sylow :-subgroup of W8 is :PÐ7Ñ œ $ :+5 Q Ð5Ñ 7

5œ"

In particular, the order of a Sylow :-subgroup of W:5 is :Q Ð5Ñ

Proof. The number of factors in 8x œ " † #â8 that are multiples of : is g8Î:h where gBh is the floor of B. Among these g8Î:h factors, there are g8Î:# h factors that are multiples of :. Thus, PÐ7Ñ œ g8Î:h  ¨8Î:# ©  â  g8Î:7 h

Using the base-: expansion of 8, we can write this as PÐ7Ñ œ Ð+"  +# :  â  +7 : 7" Ñ  Ð+#  +$ :  â  +7 : 7# Ñ  â  +7 œ +"  +# Ð:  "Ñ  â  +7 Ð: 7"  â  "Ñ œ +" Q Ð"Ñ  +# Q Ð#Ñ  â  +7 Q Ð7Ñ … Let us first determine the Sylow : -subgroups of the symmetric groups W:8 . Since the order of such a Sylow :-subgroup is :Q Ð5Ñ , all we need to do is find a subgroup of W:5 of size :Q Ð5Ñ œ :: Since

5

:5" â"

Sylow Theory

259

Q Ð5  "Ñ œ :Q Ð5Ñ  " it follows that

:Q Ð5"Ñ œ ˆ:Q Ð5Ñ ‰ † : :

and this puts us in mind of the wreath product, since if H is a finite group and kUk œ kHk œ :, then kH ›› H Uk œ kHk: † :

Now, the cyclic group G: acts faithfully on itself by left translation and so [" ³ G: ä WG: ¸ W: and since the Sylow :-subgroups of W: have order :, they are isomorphic to G: . Since G: acts faithfully on itself by left translation, the regular wreath product [# œ G: ›› < G: acts faithfully on G: ‚ G: œ G:# , that is, [# ä WG:# ¸ W:# and since 9Ð[# Ñ œ :: † : œ : :" œ : Q Ð#Ñ it follows that the Sylow :-subgroups of W:# are isomorphic to [# . Now, [# acts faithfully on G:# and G: acts faithfully on itself and so [$ œ ÐG: ›› < G: Ñ ›› < G: acts faithfully on G:$ , that is, [$ ä WG:$ ¸ W:$ and since 9Ð[$ Ñ œ Ð::" Ñ: † : œ : :

#

:"

œ : Q Ð$Ñ

it follows that the Sylow :-subgroups of W:$ are isomorphic to [$ . In general, if [8" acts faithfully on G:8" , then the 8-fold regular wreath product [8 œ [8" ›› < G: œ ÐÐG: ›› < G: Ñ ›› < G: Ñâ ›› < G: acts faithfully on G:8 and so [8 ä W:8 . Moreover, since

260

Fundamentals of Group Theory 9Ð[8 Ñ œ k[8" k: † : œ :Q Ð8"Ñ:" œ :Q Ð8Ñ

it follows that the Sylow :-subgroups of W:8 are isomorphic to [8 . To determine the nature of the :-Sylow subgroups of W8 , note that if c œ ÖF" ß á ß F7 × is a partition of M8 œ Ö"ß á ß 8× and if 53 is a permutation of F3 , then the map 5 œ 5" ‚ â ‚ 57 defined by 5Ð5Ñ œ 54 Ð5Ñ if 5 − F4 , is a permutation of M8 . It follows that if kF5 k œ 85 , then X œ W8" } â } W87 is isomorphic to a subgroup of W8 . We would like to find a partition c of M8 œ Ö"ß á ß 8× whose block sizes are powers of :. We can do this from the base-: representation of 8: 8 œ +!  + " :  â  + 7 : 7 by letting c œ ÖF5ß4 ± 4 œ "ß á ß +5 × be any partition of M8 consisting of +5 blocks of size :5 . Each symmetric group WF5ß4 is isomorphic to a subgroup of W8 where we simply let 5 − WF5ß4 be the identity on M8 Ï F5ß4 . If ]5ß4 is a Sylow : -subgroup of WF5ß4 , then the direct product ] œ } ]5ß4 5ß4

is isomorphic to a subgroup of W8 of order $ :+5 Q Ð5Ñ 7

5œ"

and since ] has the correct order, it is isomorphic to a Sylow : -subgroup of W8 . Theorem 8.34 Let : be a prime dividing 8x and let 8 œ +!  + " :  â  + 7 : 7 be the base-: representation of 8, that is, ! Ÿ +5  : .

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261

1) The Sylow :-subgroups of W:8 are isomorphic to the 8-fold regular wreath product [8 œ ÐÐG: ›› < G: Ñ ›› < G: Ñâ ›› < G: 2) The Sylow :-subgroups of W8 are isomorphic to a direct product ] œ Ð]" Ñ+! } Ð]: Ñ+" } â } Ð]:7 Ñ+7 where ]:5 is a Sylow :-subgroup of W:5 .…

Exercises 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

Prove that if L is a normal :-subgroup of a finite group K, then L is contained in every Sylow :-subgroup of K. Let the order of K be a product :;< of three distinct primes, with :  ;  <. Show that if one of 8; or 8< is equal to ", then K has a normal subgroup of order ;<. Find all Sylow subgroups of W$ . What are 8# and 8$ ? Find all Sylow subgroups of E% . What are 8# and 8$ ? Show that E& has no subgroup of index %. Show that no group of order &' is simple. Let 9ÐKÑ œ :7 Ð%:  "Ñ for 7   ". Show that 8: œ " or 8: œ %:  " or else : œ # and 8: œ $. Show that there are no simple groups of order :# ; , where : and ; are primes. Let 8 œ :5 7 with Ð7ß :Ñ œ ". Show that if :5 y± Ð7  "Ñx, then there is no simple group of order 8. This holds if 5 is “sufficiently large” relative to 9ÐKÑÎ:5 . Show that there is no simple group of order )&). Show that there is no simple group of order $#%. Show that there is no simple group of order $$*$. Show that there is no simple group of order %!*&. Show that any group of order &'" is abelian. Let 9ÐKÑ œ '!. Prove that if 8&  ", then K is simple. Hint: Use the fact that groups of order "&, #! and $! have a normal subgroup of order &. Let 5   $ be an odd integer. Show that every Sylow : -subgroup of the dihedral group H#5 is cyclic. How many Sylow #-subgroups does E& contain? Let R ü K and suppose that W ü R is a normal Sylow :-subgroup of R . Prove that W is normal in K. Let K be a finite group. Prove that K is the group generated by all of its Sylow subgroups. Let K be a finite group and let R ü K. Let W be a Sylow : -subgroup of K not contained in R . Show that WR ÎR is a Sylow :-subgroup of KÎR . Let L be a :-subgroup of a finite group K. Let +  L be a :-element. Is ØLß +Ù necessarily a : -subgroup of K? Does it help to assume that L ü K?

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22. A subgroup L of a group K is abnormal if + − ØLß L + Ù for all + − K. Prove that if O ü K and W − Syl: ÐOÑ, then RK ÐWÑ is abnormal in K. In particular, the normalizer of a Sylow :-subgroup of K is abnormal. 23. Let K be a finite group and let Lß O Ÿ K . a) Suppose that L ü K and O ü K. Show that if W is a Sylow :subgroup of K, then W œ ÐW ∩ LÑÐW ∩ OÑ. b) Show that if we drop the condition that both subgroups be normal, then the conclusion of the previous part may fail. c) Assume that L ü K. Show that for each prime : ± 9ÐKÑ, there is some Sylow :-subgroup W for which W ∩ L is a Sylow :-subgroup of L , W ∩ O is a Sylow :-subgroup of O and W ∩ L ∩ O is a Sylow :subgroup of L ∩ O . d) Assume that L ü K. Show that for each prime : ± 9ÐKÑ, there is some Sylow :-subgroup W for which W œ ÐW ∩ LÑÐW ∩ OÑ. 24. (S. Abhyankar) Let K be a finite group and let : be a prime dividing 9ÐKÑ. Let :ÐKÑ be the subgroup of K generated by the union of the Sylow :subgroups of K. Show that :ÐKÑ ü K . A finite group K is a quasi :-group if :ÐKÑ œ K. Prove that the following are equivalent: a) K is a quasi :-group. b) K is generated by all of its :-elements. c) K has no nontrivial quotient group whose order is relatively prime to :.

Chapter 9

The Classification Problem for Groups

The Classification Problem for Groups One of the most important outstanding problems of group theory is the problem of classifying all groups up to isomorphism. This is the classification problem for groups. More precisely, isomorphism of groups is an equivalence relation. Therefore, a set of canonical forms or a complete invariant constitutes a theoretical solution to the classification problem for groups. Of course, it may not be a practical solution unless some form of “algorithm” is available for determining the canonical form or invariant of any group. The classification problem for groups is unsolved and seems to be exceedingly difficult. It is even beyond present day ability to classify all finite groups. All finite abelian groups have been classified. Indeed, we shall see in a later chapter that a finite group is abelian if and only if it is the direct sum of cyclic groups of prime power orders. Moreover, the multiset of prime powers (which need not be distinct) is a complete invariant for isomorphism. All finite simple groups seem to have been classified. We will elaborate on this in more detail below.

The Classification Problem for Finite Simple Groups The classification problem for finite simple groups is generally believed by experts in the field to have been solved. The classification theorem is the following: Up to isomorphism, a finite simple group is one of the following: 1) A cyclic group G: of prime order. 2) An alternating group E8 for 8   &. 3) A classical linear group. 4) An exceptional or twisted group of Lie type. 5) A sporadic simple group (these include Mathieu groups, Janko groups, Conway groups, Fischer groups, Monster groups and more).

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The effort to solve this problem spanned the years from roughly 1950 to 1980 and involves something on the order of 15,000 pages of mathematics produced by a variety of researchers, some of which is as yet unpublished. As a result, a second effort, led by three group theorists: Daniel Gorenstein, Richard Lyons, and Ron Solomon has been underway to collect this massive effort into a single source, which now spans five volumes and will, when finished, treat most (but not all) of the overall project. The shear massiveness of this work has prompted some to believe that it is too soon to say categorically that the classification theorem as it is currently formulated is indeed a theorem. This viewpoint is further supported by the fact that, in the ensuing years since 1980 several gaps, some of which were quite serious, have appeared. Fortunately, all of the known gaps have since been filled. As an example, Michael Aschbacher [2] writes in his 2004 article The Status of the Classification of Finite Simple Groups as follows: I have described the Classification as a theorem, and at this time I believe that to be true. Twenty years ago I would also have described the Classification as a theorem. On the other hand, ten years ago, while I often referred to the Classification as a theorem, I knew formally that that was not the case, since experts had by then become aware that a significant part of the proof had not been completely worked out and written down. More precisely, the socalled “quasithin groups” were not dealt with adequately in the original proof. Steve Smith and I worked for seven years, eventually classifying the quasithin groups and closing this gap in the proof of the Classification Theorem. We completed the write-up of our theorem last year; it will be published (probably in 2004) by the AMS. Let us take a very broad look at the approach taken to solve the classification problem for finite simple groups. The abelian case is easily settled, so we will concentrate our remarks on finite nonabelian simple groups. We have already mentioned the famous result of Feit–Thompson (whose proof runs about 255 pages itself) that says that every nonabelian finite simple group has even order. This implies that any nonabelian finite simple group K contains an involution , . Moreover, the centralizer GÐ,Ñ is a nontrivial proper subgroup of K, since the center of K is trivial. Thus, every nonabelian finite simple group has a nontrivial proper involution centralizer GÐ,Ñ. At least this gives us a starting point for an investigation: Perhaps one can relate the structure of the whole group K to that of GÐ,Ñ.

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265

Indeed, we will show that if K is a nonabelian finite simple group with involution centralizer GÐ,Ñ, then 9ÐKÑ ¹ Œ

kGÐ,Ñk  " x #

The significance of this result is that kGÐ,Ñk, and therefore the right-hand side above, does not depend on the structure of K nor on the nature of GÐ,Ñ beyond its size. Thus, for a given even number ; , there are only a finite number of possible orders of groups that have an involution centralizer of size ; . But for each finite order, there are only finitely many isomorphism classes of groups of that order, because there are only finitely many multiplication tables of that size. It follows that there are only finitely many isomorphism classes of nonabelian finite simple groups that have an involution centralizer of size ; . Thus, the size of an involution centralizer does at least restrict the number of possible isomorphism classes of its parent groups. This raises the question of whether more details about the structure of an involution centralizer (and related substructures) might do more than just restrict the number of isomorphism classes for its parent. Indeed, in 1954, Richard Brauer proposed that for a finite nonabelian simple group K with involution centralizer GÐ,Ñ, the possible isomorphism classes of K are determined by the isomorphism class of GÐ,Ñ. Moreover, during the period of 1950–1965, Brauer and others developed methods for determining the isomorphism classes of all finite simple groups that have an involution centralizer isomorphic to a given group L . However, involution centralizers alone prove not to be sufficient to solve the classification problem for nonabelian finite simple groups. Note that an involution centralizer GÐ,Ñ is also the normalizer RK ÐØ,ÙÑ of the #subgroup Ø,Ù. More generally, the normalizer RK ÐLÑ of a : -subgroup L of a group K is called a :-local subgroup of K. Thus, an involution centralizer is a special type of #-local subgroup of K. The search for nonisomorphic nonabelian finite simple groups involves looking at the entire :-local structure of a group and can be roughly described as follows: 1) If the current list of nonisomorphic nonabelian finite simple groups is not complete, let K be a minimal counterexample. Thus, any proper subgroup of K is on the list. 2) Show that the :-local structure of K resembles that of a simple group W that is already on the list. 3) Use this resemblance to show that K is isomorphic to W . If not, then perhaps K must be added to the list. So let us proceed to the promised theorem.

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Involutions Let K be an even-order group whose set \ of involutions has size 8   " and let \ w œ \ ∪ Ö"× If = is an involution, then for any involution >, we have > œ =Ð=>Ñ œ =B where B œ => has the property that B= œ B" This property of B is very important. Definition Let K be a group. 1) An element B − K is real by + − K if B+ œ B" Also, B is real if it is real by some element +. Let e be the set of real elements of K. 2) An element B − K is strongly real by = if B= œ B" where = is an involution. Also, B is strongly real if it is strongly real by some involution =. Let f denote the set of strongly real elements of K.… Thus, every pair of involutions is related by a strongly real element. It is not hard to see that \ , f and e are each closed under conjugation and so each set is a union of conjugacy classes. Associated with the equation B+ œ B" are some important sets. Definition Let K be a group. 1) For B − K, let G w ÐBÑ œ Ö+ − K ± B+ œ B" × be the set of all elements by which B is real. The extended centralizer of B − K is G ‡ ÐBÑ œ GÐBÑ ∪ G w ÐBÑ where GÐBÑ is the centralizer of B.

The Classification Property for Groups 2

267

a) For B − f , let EÐBÑ œ Ö= − \ ± B= œ B" × be the set of involutions by which B is strongly real. b) For = − \ , let fÐ=Ñ œ ÖB − K ± B= œ B" × be the set of strongly real elements by the involution =.…

Let us take a look at these sets. Note first that B − K is real if and only if G w ÐBÑ is nonempty and B − K is strongly real if and only if EÐBÑ is nonempty. G ‡ ÐBÑ If B − e and +ß , − G w ÐBÑ, then B+ œ B" œ B, and so +" , − GÐBÑ. Thus, G w ÐBÑ is a coset of GÐBÑ, G w ÐBÑ œ +GÐBÑ and so

kG w ÐBÑk œ kGÐBÑk

Thus, for B − e and + − G w ÐBÑ, G ‡ ÐBÑ œ GÐBÑ ∪ +GÐBÑ œ œ

GÐBÑ GÐBÑ “ +GÐBÑ

if B − \ w if B − e Ï \ w

where if B − e Ï \ w , then ÐG ‡ ÐBÑ À GÐBÑÑ œ # EÐBÑ If B − f , then EÐBÑ © G w ÐBÑ and so

kEÐBÑk Ÿ kGÐBÑk

But we can do a bit better in some cases. Since EÐ"Ñ œ \ , we have kEÐ"Ñk œ 8

and if B − \ , then EÐBÑ © GÐBÑ Ï Ö"× and so

kEÐBÑk Ÿ kGÐBÑk  "

Also, for any 1 − K, it is easy to see that EÐB1 Ñ œ EÐBÑ1

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Fundamentals of Group Theory

and so

kEÐB1 Ñk œ kEÐBÑk

that is, kEÐBÑk is constant on conjugacy classes of K. WÐ=Ñ If = − \ , then we have seen that \ w © =f Ð=Ñ Conversely, if B − f Ð=Ñ for = − \ , then Ð=BÑ# œ =B=B œ B" B œ " and so > œ =B − \ w . Hence, \ w œ =f Ð=Ñ and so

kf Ð=Ñk œ k=f Ð=Ñk œ 8  "

The Fundamental Relation Now we can count the size of the set Y œ ÖÐ=ß BÑ − \ ‚ f ± B= œ B" × in two ways. From the point of view of an = − \ , kY k œ

=−\

kf Ð=Ñk œ 8Ð8  "Ñ

From the point of view of an B − f ,

kY k œ

B−f

kEÐBÑk

Thus, 8#  8 œ B−f

kEÐBÑk

(9.1)

To split up the sum on the right, we choose a system of distinct representatives (SDR) Q œ ÖB! ß B" ß á ß B> × for the conjugacy classes of K, where 1) ÖB! œ "× 2) ÖB" ß á ß B? × is an SDR for the conjugacy classes in \

The Classification Property for Groups

269

3) ÖB?" ß á ß B@ × is an SDR for the conjugacy classes in f Ï \ w Then since kEÐBÑk is constant on conjugacy ¸BK ¸ œ ÐK À GÐBÑÑ, equation (9.1) can be written @

8#  8 œ 3œ!

classes

and

since

kEÐB3 ÑkÐK À GÐB3 ÑÑ

Splitting this sum further gives ?

8#  8 œ 8  3œ" ?

Ÿ8 3œ"

kEÐB3 ÑkÐK À GÐB3 ÑÑ 

œ 8  @kKk  and so

3œ?"

kEÐB3 ÑkÐK À GÐB3 ÑÑ

ÐkGÐB3 Ñk  "ÑÐK À GÐB3 ÑÑ 

Ÿ 8  ?kKk  œ @kKk

@

? 3œ" ? 3œ"

@ 3œ?"

kGÐB3ÑkÐK À GÐB3ÑÑ

ÐK À GÐB3 ÑÑ  Ð@  ?ÑkKk

¸BK ¸ 3

8#  8 Ÿ @kK k

(9.2)

To get further estimates, note that 8 œ k\ k œ

?

ÐK À GÐB3 ÑÑ

(9.3)

3œ"

Now, if we assume that the center ^ÐKÑ of K has odd order, then it contains no involutions and so GÐB3 Ñ  K for B3 − \ . Hence, if 7 is the smallest index among all proper subgroups of K, we have 8 œ k\ k   7?

We can make a similar estimate for kf k œ "  8 

@

ÐK À GÐB3 ÑÑ 3œ?"

and since B3 − f Ï \ w , the terms in the final sum satisfy ÐK À GÐB3 ÑÑ œ ÐK À G ‡ ÐB3 ÑÑÐG ‡ ÐB3Ñ À GÐB3ÑÑ œ #ÐK À G ‡ÐB 3ÑÑ Therefore, if we assume that G ‡ ÐB3 Ñ is also proper in K , then

(9.4)

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Fundamentals of Group Theory

ÐK À GÐB3 ÑÑ   #7 and so

kf k   "  8  #7Ð@  ?Ñ

(9.5)

The condition that G ‡ ÐB3 Ñ is proper in K is a bit awkward, but is satisfied if we assume that K has no subgroups of index #. The inequalities (9.4) and (9.5) together imply that @Ÿ

kf k  "  8 #7

and so (9.2) implies that 8#  8 Ÿ

kf k  "  8 kKk #7

Some elementary algebra, using the fact that kf k Ÿ kKk, gives 7Ÿ

" kKk kK k " Œ Œ # 8 8

œŒ

kKkÎ8  " #

We have proved a key theorem. Theorem 9.6 (R. Brauer and K. A. Fowler [4], 1955) Let K be a group of even order with exactly 8   " involutions. Assume that ^ÐKÑ has odd order. Then either K has a subgroup of index # (which must be normal) or K has a proper subgroup L with ÐK À LÑ Ÿ Œ

kKkÎ8  " #

…

Equation (9.3) implies that for any involution , , which we can assume is B" , we have 8 œ kK k that is,

? 3œ"

" "   kGÐB3 Ñk kGÐ,Ñk

kKk Ÿ kGÐ,Ñk 8

Hence, if K fails to have a (normal) subgroup of index #, then it has a subgroup L for which

The Classification Property for Groups

ÐK À LÑ Ÿ Œ

271

kGÐ,Ñk  " #

Since L is a proper subgroup of K, it follows that the normal interior L ‰ is a proper normal subgroup with ÐK À L ‰ Ñ ± ÐK À LÑx. In particular, if K is simple, then 9ÐKÑ ± ÐK À LÑx and so 9ÐKÑ º Œ

kGÐ,Ñk  " x #

Thus, we arrive at our final goal. Theorem 9.7 Let K be a finite nonabelian simple group and let , − K be an involution. Then the centralizer GÐ,Ñ is a proper subgroup of K and 9ÐKÑ º Œ

kGÐ,Ñk  " x #

…

This is the result that we promised at the beginning of this section and is as far as we propose to take our discussion of the classification problem for nonabelian finite simple groups.

Exercises 1.

2. 3. 4. 5. 6. 7.

Let K be a group. a) Under what conditions does the set W œ \ ∪ Ö"× of elements of K of exponent # form a subgroup of K? b) Under what conditions does the set W form a normal subgroup of K? c) If W is a subgroup of K, what can you say about the strongly real elements of the group? Prove that f is closed under conjugation. Prove that if a finite group K has a nontrivial real element, then K has even order. Find the real elements in the symmetric group W8 . Find the strongly real elements. In the alternating group E8 , show that any permutation that is a product of disjoint cycles of length congruent to " modulo % is real. Find the real elements of the dihedral group H#8 . Find the strongly real elements. Find the real elements of the quaternion group U. Find the strongly real elements.

Chapter 10

Finiteness Conditions

There are many forms of finiteness that a group can possess, the most obvious of which is being a finite set. However, as we have observed, chain conditions are also a form of finiteness condition. Another type of finiteness condition on a group K is the condition that K has a finite direct sum decomposition K œ H"  â  H 8 that cannot be further refined by decomposing any of the factors H3 into a direct sum; that is, for which each H3 is indecomposable. In this chapter, we explore these finiteness conditions. First, however, we generalize the notion of a group.

Groups with Operators As we have seen, a group K has several important families of subgroups, in particular, the families of all subgroups, all normal subgroups, all characteristic subgroups and all fully-invariant subgroups. Each of these families can be characterized as being the family of all subgroups that are invariant under a certain subset of EndÐKÑ. In particular, a subgroup L of K is 1) normal if and only if it is invariant under InnÐKÑ, 2) characteristic if and only if it is invariant under AutÐKÑ, 3) fully invariant if and only if it is invariant under EndÐKÑ. We can also say that the subgroups of K are invariant under the empty subset of EndÐKÑ. This point of view leads us to define the concept of groups with operators, which will include all of these special cases. Intuitively speaking, a group with operators is a group K with a distinguished family X of endomorphisms of K. Rather than associate a subgroup of EndÐKÑ with K directly, we use a function 0 À H Ä EndÐKÑ from a set H into EndÐKÑ. Here is the formal definition.

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Definition Let H be a set. An H-group is a pair ÐKß 0 À H Ä EndÐKÑÑ where K is a group and 0 À H Ä EndÐKÑ is a function. It is customary to denote the endomorphism 0 Ð=Ñ simply by = and thus write 0 Ð=Ñ+ as =+ (some authors write += ). An H-group is called a group with operators and H is called the operator domain. Let K be an H-group. 1) An H-subgroup L of K is an H-invariant subgroup L of K. We use the notations L Ÿ H K and L ü H K to denote an H-subgroup and a normal Hsubgroup of K, respectively. We also use the notations H-subÐKÑ and

H-norÐKÑ

to denote the set of all H-subgroups of K and the set of all normal Hsubgroups of K, respectively. 2) If K and L are H-groups, an H-homomorphism from K to L is a homomorphism 5À K Ä L that is compatible with the group operators, that is, 5Ð=+Ñ œ =Ð5+Ñ for all + − K. A bijective H-homomorphism is an H-isomorphism, and similarly for the other types of homomorphisms. We write 5À K ¸ H L to denote an H-isomorphism from K to L . The existence of an H-isomorphism from K to L is denoted by K ¸ H L . 3) If L − H-norÐKÑ, then the H-quotient group (or H-factor group) is the quotient group KÎL with operators = − H defined by =Ð+LÑ œ Ð=+ÑL for all + − K, that is, defined so that the canonical projection 1L is an Hhomomorphism.… Let ÐKß 0 À H Ä EndÐKÑÑ be an H-group. Then a subgroup L Ÿ K is an Hsubgroup of K if and only if the restricted operators HlL œ Ö0 Ð=ÑlL ± = − H× are operators on L , that is, if and only if L is an HlL -group. We will always think of an H-subgroup L of K as an HlL -group, although we will use notation such as O Ÿ H L in place of O Ÿ HlL L . Thus, H-subgroupness is transitive, that is, L Ÿ H Kß

O ŸH L

Ê

O ŸH K

Note that an H-subgroup L of K may be an operator group in other related ways. To illustrate, if H œ InnÐKÑ, then L is an HlL -group as well as an operator group under its own family InnÐLÑ of inner automorphisms. But in

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275

general, InnÐLÑ is a proper subset of InnÐKÑlL , since conjugation by + − K Ï L need not be an inner automorphism of L . If H is the empty set, then an H-group is nothing more than an ordinary group and it is customary to drop the prefix “H-”. Also, in the most important cases, H is a subset of EndÐKÑ and 0 À H Ä EndÐKÑ is the inclusion map, in which case 0 is suppressed. This applies to the cases H œ InnÐKÑ, H œ AutÐKÑ and H œ EndÐKÑ. Example 10.1 Let Z be a vector space over a field J . Each α − J defines an endomorphism of the abelian group Z by scalar multiplication. Thus, a vector space over J is a group with operators J . An J -subgroup is a subspace and an J -homomorphism is a linear transformation.…

The Lattice of H-Subgroups of an H-Group Let K be an H-group. Then the intersection and the join of any family Y œ ÖL3 ± 3 − M× of H-subgroups of K is also an H-subgroup of K. Hence, the meet and join in H-subÐKÑ is the same as the meet and join in subÐKÑ. In other words, H-subÐKÑ is a complete sublattice of subÐKÑ. We leave it to the reader to show that the H-subgroup Ø\ÙH generated by a nonempty subset \ of K is Ø\ÙH œ Ø=B ± B − \ß = − HÙ An H-subgroup L of an H-group K is finitely H-generated if L œ Ø\ÙH for some finite set \ . This generalizes the normal closure of a subset \ of K.

The H-Isomorphism and H-Correspondence Theorems The concept of universality given in Theorem 4.5 and the consequent isomorphism theorems have direct generalizations to groups with operators. Let K be a H-group and let O Ÿ H K. Let YH ÐKà OÑ be the family of all pairs ÐLß 5À K Ä LÑ, where 5À K Ä L is an H-homomorphism and O © kerÐ5Ñ. Then the pair ÐKÎOß 1O À K Ä KÎOÑ is universal in YH ÐKà OÑ, in the sense that for any pair ÐLß 5À K Ä LÑ in YH ÐKà OÑ, there is a unique mediating H-homomorphism 7 À W Ä L for which 7 ‰ 1O œ 5 To see this, note that Theorem 4.5 guarantees the existence of a mediating homomorphism 7 . But if 1O and 5 are H-homomorphisms, then for any = − H and + − K, 7 Ð=Ð+OÑÑ œ 7 ÐÐ=+ÑOÑ œ 5Ð=+Ñ œ =Ð5+Ñ œ =7 Ð+OÑ and so 7 is also an H-homomorphism. Also, H-universality enjoys the same

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uniqueness up to isomorphism as ordinary universality (H œ g). The Hisomorphism theorems now follow in the same manner as before. Theorem 10.2 (The H-isomorphism theorems) Let K be an H-group. 1) (First H-isomorphism theorem) Every H-homomorphism 5À K Ä L induces an H-embedding 5À KÎkerÐ5Ñ ä L defined by 5Ð1kerÐ5ÑÑ œ 5Ð1Ñ and so K ¸ H imÐ5Ñ kerÐ5Ñ 2) (Second H-isomorphism theorem) If Lß O − H-subÐKÑ with O ü K, then L ∩ O − H-norÐLÑ and LO L ¸H O L ∩O 3) (Third H-isomorphism theorem) If L Ÿ O Ÿ K with Lß O − H-norÐKÑ, then OÎL − H-norÐKÎLÑ and K O K „ ¸H L L O 4) (The H-correspondence theorem) Let R − H-norÐKÑ. If H-subÐR à KÑ denotes the lattice of all H-subgroups of K that contain R , then the map 1À subÐR à KÑ Ä subÐKÎR Ñ defined by 1ÐLÑ œ LÎR preserves H-invariance in both directions and so maps H-subÐR à KÑ bijectively onto H-subÐKÎR Ñ.…

H-Series and H-Subnormality We can now generalize the notion of series and subnormality to groups with operators. This will provide considerable time savings in our later work. Definition Let K be an H-group. 1) An H-series in K is a series Z À K! ü K" ü â ü K8 in which every term is an H-subgroup of K . 2) A refinement of an H-series Z is an H-series [ obtained from Z by including zero or more additional H-subgroups between the endpoints. A proper refinement is a refinement that includes at least one new subgroup.…

Finiteness Conditions

277

We review the usual suspects in the context of H-series: a) If H œ g, an H-series is just a series. b) If H œ InnÐKÑ, an H-series is a normal series. c) If H œ AutÐKÑ, an H-series is a characteristic series. d) If H œ EndÐKÑ, an H-series is a fully invariant series. Of course, if ÖK3 × is a nonproper H-series for K, we can dedup the series by removing any duplicate subgroups to obtain a proper series. Definition Let K be an H-group. Two equal-length H-series Z À K! ü K" ü â ü K8" ü K8 and [À L! ü L" ü â ü L8" ü L8 in K with common endpoints K! œ L! and K8 œ L8 are H-isomorphic (also called H-equivalent) if there is a bijection 0 of the index set Ö!ß á ß 8  "× for which K3" ÎK3 ¸ H L0 Ð3Ñ" ÎL0 Ð3Ñ

…

As usual, when H œ g, we use the term isomorphic. Thus, for example, the series Ö"× – G: – ÐG:  G; Ñ and Ö"× – G; – ÐG:  G; Ñ are isomorphic.

H-Subnormality H-subnormality of a subgroup L Ÿ H K requires not just that L be subnormal and H-invariant, but that the entire series that witnesses subnormality be an Hseries. Definition Let K be an H-group. An H-subgroup L of K is H-subnormal in K , denoted by L ü ü H K, if there is an H-series from L to K. We use the notations subnH ÐKÑ

and subnH ÐR à KÑ

to denote the family of all H-subnormal subgroups of K and the family of all Hsubnormal subgroups of K that contain R , respectively.… We can now generalize Theorem 4.24.

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Fundamentals of Group Theory

Theorem 10.3 Let K be an H-group and let Lß O − H-subÐKÑ. 1) (Transitivity) L ü ü HO

and O ü ü H K

Ê

L ü ü HK

2) (Intersection) If P − H-subÐKÑ, then L ü ü HO

Ê

L ∩ P ü ü HO ∩ P

In particular, L Ÿ O Ÿ Kß

L ü ü HK

Ê

L ü ü HO

and L ü ü H Kß

O ü ü HK

Ê

L ∩ O ü ü HK

3) (Normal lifting) If R − H-norÐKÑ, then L ü ü HO

Ê

LR ü ü H OR

4) (Quotient/unquotient) If R − H-norÐOÑ and R Ÿ L Ÿ O , then L ü ü HO

Í

LÎR ü ü H OÎR

…

Composition Series If K! ß K8 Ÿ H K, then refinement is a partial order on the set of all proper Hseries from K! to K8 (assuming that this set is nonempty). Maximal proper Hseries are particularly important. Definition Let K be an H-group and let K! ß K8 Ÿ H K . An H-composition series from K! to K8 is a proper H-series K! – K" – â – K8 that is maximal in the family of all proper H-series from K! to K8 , under refinement. If there is an H-composition series from K! to K8 , we will write bCompSerH ÐK! à K8 Ñ

or bCompSerH ÐK8 Ñ when K! œ Ö"×

The factor groups of an H-composition series are called H-composition factors. 1) A maximal series (H œ g) is simply called a composition series and the factor groups are called composition factors. 2) A maximal normal series (H œ InnÐKÑ) in K is called a chief series or principal series and the factor groups are called chief factors or principal factors.… When the endpoints of a series are clear from the description of the series, we will drop the “from-to” terminology. To characterize maximal series, we use the following concept.

Finiteness Conditions

279

Definition A nontrivial H-group K is H-simple if K has no nontrivial proper normal H-subgroups.… The H-correspondence theorem implies the following. Theorem 10.4 A proper H-series is an H-composition series if and only if its factor groups are H-simple.… Thus, a series ZÀ K! – K" – â – K8 in K is a composition series if and only if its factor groups K5" ÎK5 are simple and Z is a chief series if and only if each factor group K5" ÎK5 is a minimal normal subgroup of KÎK5 . It is clear from Theorem 10.4 that any H-series that is H-isomorphic to an Hcomposition series is also an H-composition series. Also, if we remove an endpoint from an H-composition series, the result is also an H-composition series (with different endpoints, of course). Finally, if K! – â – K5

and K5 – â – K8

are H-composition series in K, then so is their concatenation K! – â – K8

The Extension Problem An extension of a pair ÐR ß UÑ of groups is a group K that has a normal subgroup R w isomorphic to R and for which KÎR w is isomorphic to U. The extension problem for the pair ÐR ß UÑ is the problem of determining (up to isomorphism) all possible extensions of ÐR ß UÑ. Note that any external semidirect product K œ R z ) U is an extension of ÐR ß UÑ. However, semidirect products alone do not solve the extension problem. For example, ™ is an extension of Ð#™ß ™# Ñ but ™ is not a semidirect of any of its nontrivial proper subgroups. The importance of the extension problem can be clearly seen in the light of composition series. Suppose that we can solve the extension problem and that we can determine (up to isomorphism) all simple groups. The simple groups are precisely the groups that have a composition series of length ": Ö"× œ K! – K" Next, for each K" , we solve the extension problem for all pairs of the form ÐK" ß L" Ñ, where L" ranges over the simple groups. The solutions K# are precisely the groups that have composition series of length #:

280

Fundamentals of Group Theory

K! – K" – K# where K# ÎK" ¸ L" . Continuing to extend by all possible simple groups produces all possible groups that have composition series, and this includes all finite groups. Thus, in particular, if we can solve the extension problem and if we can determine all finite simple groups, we can determine all finite groups. Unfortunately, a practical solution to the extension problem does not exist at this time.

The Zassenhaus Lemma and the Schreier Refinement Theorem Let K be an H-group. Our next goal is to show that any two H-series in K with the same endpoints have refinements that are H-isomorphic. This result is called the Schreier refinement theorem and has two extremely important consequences, as we will see. First, let us recall that the projection ÐE ü FÑ Ä ÐL ü OÑ of E ü F into L ü O is the extension LÐE ∩ OÑ ü LÐF ∩ OÑ Moreover, when E, F , L and O are H-subgroups, then so are LÐE ∩ OÑ and LÐF ∩ OÑ and the Zassenhaus lemma (Theorem 4.12) generalizes directly to H-subgroups. Theorem 10.5 (Zassenhaus lemma [37], 1934) Let K be an H-group and let EüF

and

LüO

where Eß Fß Lß O − H-subÐKÑ. Then the reverse projections ÐE ü FÑ Ä ÐL ü OÑ and

ÐL ü OÑ Ä ÐE ü FÑ

have H-isomorphic factor groups, that is, LÐF ∩ OÑ EÐO ∩ FÑ ¸H LÐE ∩ OÑ EÐL ∩ FÑ

…

Now we can prove the Schreier refinement theorem. Let L Ÿ O be Hsubgroups of an H-group K and consider a pair of H-series from L to O : Z À L œ K! ü K " ü â ü K 8 œ O and [À L œ L! ü L" ü â ü L7 œ O Projecting each of the 7 steps of [ into each of the 8 steps of Z creates a new H-series with 78 steps, some of which may be trivial. In view of the preceeding

Finiteness Conditions

281

remarks, the new H-series is a refinement of Z . Similarly, projecting each of the 8 steps of K into each of the 7 steps of [ creates a new H-series with 78 steps. Moreover, since the two sets of projections consist of inverse pairs, the Zassenhaus lemma implies that the resulting series are H-isomorphic. Theorem 10.6 (Schreier refinement theorem, Schreier [29], 1928) Let K be an H-group. Then any two H-series in K with the same endpoints have Hisomorphic refinements.…

Consequences of the Schreier Refinement Theorem The Schreier refinement theorem has two very important consequences. First, suppose that there is an H-composition series V in K from L to O . Then any proper H-series [À L! – L" – â – L8 with H-subnormal endpoints between L and O can be refined to an Hcomposition series. To see this, note that since L! and L8 are H-subnormal, the series [ can be expanded to an H-series ^ from L to O and the Schreier refinement theorem implies that V and ^ have H-isomorphic refinements to proper series Vw and ^w , respectively. But Vw œ V is an H-composition series and therefore so is ^w , which contains a refinement of [. The second consequence of the Schreier refinement theorem is that any two Hcomposition series with the same endpoints are H-isomorphic. Theorem 10.7 Let K be an H-group and let L Ÿ O be H-subgroups of K. 1) Suppose that there is an H-composition series from L to O of length 8. If L! Ÿ L8 are H-subnormal subgroups of K between L and O , then any Hseries from L! to L8 can be refined to an H-composition series and so has length at most 8. 2) (The Jordan–Hölder Theorem) Every two H-composition series from L to O are H-isomorphic. In particular, they have the same length.… The Jordan–Hölder Theorem allows us to make the following definition. Definition Let K be an H-group and let L and O be H-subgroups of K . If bCompSerH ÐLà OÑ, then the H-composition distance .ÐLß OÑ from L to O is the length of any H-composition series from L to O . The distance .ÐOÑ œ .ÐÖ"×ß OÑ is called the H-composition length of O . For chief series, the terms chief distance and chief length are also employed.… Of course, the composition distance is positive definite (when it is defined), that is, if bCompSerH ÐL à OÑ, then .ÐLß OÑ   ! and .ÐLß OÑ œ !

Í

L œO

282

Fundamentals of Group Theory

The Existence of H-Composition Series Any finite group has an H-composition series. For abelian groups, composition series and chief series coincide and an abelian group has a composition series if and only if it is finite, since each factor group must have prime order. Thus, not all groups have composition or chief series. The existence of an H-composition series between H-subnormal subgroups L and O of an H-group K can be characterized in terms of chain conditions on Hsubnormal subgroups. If bCompSerH ÐLà OÑ, then any H-series from L to O has length at most .ÐLß OÑ. Hence, Theorem 1.5 gives the following. Theorem 10.8 Let K be an H-group and let Lß O − subnH ÐKÑ. Then bCompSerH ÐLà OÑ

subnH ÐLà OÑ has BCC

Í

…

Proof of the following is left to the reader. Theorem 10.9 Let K be an H-group. 1) (Subgroup) If O − subnH ÐKÑ and L Ÿ O , then bCompSerH ÐL à KÑ

bCompSerH ÐLà OÑ and bCompSerH ÐOà KÑ

Ê

and .ÐLß KÑ œ .ÐLß OÑ  .ÐOß KÑ 2) (Quotients) If R − H-norÐKÑ, then bCompSerH ÐR à KÑ

Í

bCompSerH ÐKÎR Ñ

and .ÐR ß KÑ œ .ÐKÎR Ñ 3) (Extensions) If R − H-norÐKÑ, then bCompSerH ÐR Ñß

bCompSerH ÐKÎR Ñ

Ê

bCompSerH ÐKÑ

4) (Direct products) If K and L are H-groups, then bCompSerH ÐKÑ and bCompSerH ÐLÑ

Í

bCompSerH ÐK } LÑ

and .ÐK { LÑ œ .ÐKÑ  .ÐLÑ

The Remak Decomposition Let us recall Theorem 5.12.

…

283

Finiteness Conditions

Theorem 10.10 (Remak) If a group K has either (and therefore both) chain condition on direct summands, then K has a Remak decomposition K œ V"  â  V8 that is, each V5 is indecomposible.… The question of uniqueness of a Remak decomposition is rather more complicated than the question of existence. Recall that if K œ L"  â  L8 then the 5 th projection map 3L5 À K Ä L5 is defined by 3L5 Ð+Ñ œ Ò+ÓL5 where Ò+ÓL5 is the 5 th coordinate of +. Moreover, 3L3 is idempotent and normal as an endomorphism of K. Note also that the sum 3L3"  â  3L35 is projection onto L3"  â  L354 and so is an endomorphism of K.

The Krull–Remak–Schmidt Theorem Suppose now that K œ L"  â  L8

(10.11)

with projection maps 1L" ß á ß 1L8 and K œ O"  â  O7

(10.12)

with projection maps ,O" ß á ß ,O7 , where the factors L5 and O5 are indecomposable and 8 Ÿ 7. In searching for possible isomorphisms between the L -factors and the O factors, we recall Theorem 5.25 as it applies to the restricted projection maps 1L3 lO4 À O4 Ä L3

and

,O4 lL3 À L3 Ä O4

Namely, if Ð1L3 lO4 ÑÐ,O4 lL3 Ñ − AutÐL3 Ñ and imÐ,O4 lL3 Ñ ü O4 , then 1L3 lO4 and ,O4 lL3 are isomorphisms. Note however that imÐ,O4 lL3 Ñ ü O4 , since if 2 − L3 and 5 − O4 , then 5Ò,O4 lL3 Ð2ÑÓ5 " œ ,O4 Ð5Ñ,O4 Ð2Ñ,O4 Ð5Ñ" œ ,O4 Ð525 " Ñ − imÐ,O4 lL3 Ñ Thus, we have the following. Theorem 10.13 Let K œ L"  â  L8 œ O"  â  O7 where the factors L5 and O5 are indecomposable. If the composition

284

Fundamentals of Group Theory

α3ß4 œ Ð1L3 lO4 Ñ ‰ Ð,O4 lL3 ÑÀ L3 Ä L3 of the restricted projection maps is an automorphism of L3 , then the maps 1L3 lO4 À O4 ¸ L3

and ,O4 lL3 À L3 ¸ O4

are isomorphisms.… In attempting to show that a composition is an automorphism, we are reminded of Fitting's lemma. We have assumed that L3 is indecomposable. Also, since the restriction and composition of normal maps is normal, α3ß4 is normal and so has normal higher images. Thus, if we assume that K has BCC on normal subgroups, then Fitting's lemma implies that α3ß4 is either nilpotent or an automorphism for all 3ß 4. To see that for each 3, not all of the maps α3ß4 can be nilpotent, note that for 4 Á 5, α3ß4  α3ß5 œ Ð1L3 ,O4  1L3 ,O5 ÑlL3 œ 1L3 Ð,O4  ,O5 ÑlL3 which is an endomorphism of L3 and so the images imÐα3ß4 Ñ and imÐα3ß5 Ñ commute elementwise. Also, 1L3 ,O"  â  1L3 ,O7 œ 1L3 Ð,O"  â  ,O7 Ñ œ 1L3 and so α3ß"  â  α3ß7 œ 1L3 lL3 œ +L3 which is not nilpotent. Hence, Theorem 4.3 implies that α3ß4 is not nilpotent for some 4. It follows from Fitting's lemma that for each 3, there is a 4 for which α3ß4 − AutÐL3 Ñ and so 1L3 lO4 À O4 ¸ L3

and ,O4 lL3 À L3 ¸ O4

Now, we can make a significant improvement to this by noticing that if O4 œ L5 for some 5 Á 3, then α3ß4 œ Ð1L3 lO4 ÑÐ,O4 lL3 Ñ œ Ð1L3 lL5 ÑÐ,L5 lL3 Ñ œ ! and so if we delete from the sum 4 α3ß4 all terms indexed by a 4 for which O4 − ÖL" ß á ß L8 × Ï ÖL3 ×, the sum remains unchanged and so is not nilpotent. Hence, 1L3 lO4 À O4 ¸ L3

and ,O4 lL3 À L3 ¸ O4

for some 4 for which O4 Â ÖL" ß á ß L8 × Ï ÖL3 ×. Now suppose that after possible reindexing of the O 's, there is a 5   " for which

Finiteness Conditions

K œ O"  â  O5"  L5  â  L8

285

(10.14)

with projections ." ß á ß .8 , where -3 ³ ,O3 lL3 À L3 ¸ O3 for all " Ÿ 3 Ÿ 5  ". We may also assume that L5 Á O4 for all 4   5 or else we can replace L5 by O4 (reindexed to O5 Ñ. This certainly holds for 5 œ ". Then there is a 4   5 and we may assume that 4 œ 5 , for which 1L5 lO5 À O5 ¸ L5

and

-5 ³ ,O5 lL5 À L5 ¸ O5

Moreover, since 1L5 maps O5 isomorphically onto L5 and maps the complement LÐ5Ñ œ L"  â  L5"  L5"  â  L8 to Ö"×, it follows that LÐ5Ñ ∩ O5 œ Ö"× Thus, if we replace L5 by O5 in (10.14), the result is a direct sum K" œ O"  â  O5  L5"  â  L8 and our goal is to show that K" œ K. To this end, if .Ð5Ñ œ ."  â  .5"  .5"  â  .8 then .5  .Ð5Ñ œ +K and the map ) œ ,O5 .5  .Ð5Ñ is a normal endomorphism of K with imÐ) Ñ © O5 LÐ5Ñ . To show that ) is surjective and so K œ O5 LÐ5Ñ , it is sufficient to show that ) is injective. Now, any + − K has the form + œ B2 for B − LÐ5Ñ and 2 − L5 . But for any B − LÐ5Ñ , ) ÐBÑ œ ,O5 .5 ÐBÑ.Ð5Ñ ÐBÑ œ B and any 2 − L5 , ) Ð2Ñ œ ,O5 .5 Ð2Ñ.Ð5Ñ Ð2Ñ œ ,O5 Ð2Ñ and so ) ÐB2Ñ œ B,O5 Ð2Ñ and since LÐ5Ñ ∩ O5 œ Ö"×, it follows that )ÐB2Ñ implies B œ " and ,O5 Ð2Ñ œ ". But ,O5 lL5 À L5 ¸ O5 and so 2 œ "Þ Thus, ) is injective.

286

Fundamentals of Group Theory

It follows that K œ O"  â  O5  L5"  â  L8 for all " Ÿ 5 Ÿ 8 and -5 œ ,O5 lL5 À L5 ¸ O5 and so this holds for all 5 œ "ß á ß 8. In particular, 8 œ 7. Note also that since ,O3 ÐL3 Ñ œ O3 , the map - œ -" 1L"  â  -8 1L8 œ ,O" 1L"  â  ,O8 1L8 is a surjective normal endomorphism of K and so - − AutÐKÑ. Moreover, -ÐL5 Ñ œ O5 We can now summarize. Theorem 10.15 (The Krull–Remak–Schmidt Theorem) Let K be a group that has BCC on normal subgroups. Suppose that K œ L"  â  L8 œ O"  â  O7 where all factors L3 and O4 are indecomposable. Then 8 œ 7 and there is a reindexing of the O3 's and a normal automorphism - of K for which -À L3 ¸ O3 for all 3 œ "ß á ß 8 and for each " Ÿ 5 Ÿ 8, K œ O"  â  O5  L5"  â  L8

…

True Uniqueness The Krull–Remak–Schmidt Theorem gives uniqueness of the terms of a Remak decomposition up to isomorphism. Let us now consider the question of when a group K has an essentially unique Remak decomposition, that is, a Remak decomposition that is unique up to the order of the factors. First suppose that K œ L"  â  L8 œ O"  â  O 8 are Remak decompositions of K (where 8  "). Then the Krull–Remak– Schmidt Theorem implies that there is a normal automorphism -À K Ä K for which -L5 œ O5 (after reindexing). Hence, if L5 is invariant under every normal automorphism of K, then O5 œ -L5 Ÿ L5 and so O5 œ L5 for all 5 and K has an essentially unique Remak decomposition. By way of converse, suppose that K has an essentially unique Remak decomposition K œ L"  â  L8 with projections Ö1" ß á ß 18 ×, but that there is a normal endomorphism - of K

Finiteness Conditions

287

for which -L" Ÿ y L" . Then there is a 5  " for which 15 - Á ! on L" , that is, there is an B − L" for which " Á 15 -ÐBÑ − L5 . The set O" œ Ö2 † 15 -Ð2Ñ ± 2 − L" × is easily seen to be a normal subgroup of K and O" Á L" , since B † 15 -ÐBÑ − O" Ï L" But it is easy to see that K œ O"  L#  â  L 8 Moreover, if O" is decomposable, then there would be a Remak decomposition of K consisting of more than 8 terms, which is false. Hence, this is a Remak decomposition of K that is distinct from the previous decomposition. We have proved the following. Theorem 10.16 Let K have BCC on normal subgroups and let K œ L"  â  L8 be a Remak decomposition of K. The following are equivalent: 1) This Remak decomposition of K is essentially unique . 2) L5 is invariant under all normal endomorphisms of K. 3) L5 is invariant under all normal automorphisms of K.… If αÀ L3 Ä ^ÐKÑ is a nonzero homomorphism, then we can build a normal endomorphism -À K Ä K by specifying that -lL5 œ œ

if 5 œ 3 if 5 Á 3

α !

The map - is normal since for any + − K, 23 − L3 , 25 − L5 where 5 Á 3, -Ð23+ Ñ œ -23 œ Ð-23 Ñ+

and

-Ð25+ Ñ œ " œ Ð-25 Ñ+

Thus, if L3 is not --invariant, then Theorem 10.16 implies that the Remak decomposition of K is not unique. Conversely, suppose that the Remak decomposition of K is not unique and so there is a normal endomorphism - of K for which 14 -lL3 Á ! for some 4 Á 3. Then for 23 − L3 and 24 − L4 , 2

Ò-Ð23 ÑÓ24 œ -Ð23 4 Ñ œ -Ð23 Ñ which shows that -lL3 À L3 Ä ^ÐKÑ. Hence, 14 -lL3 À L3 Ä ^ÐL4 Ñ is a nonzero homomorphism.

288

Fundamentals of Group Theory

Theorem 10.17 Let K have BCC on normal subgroups and let K œ L"  â  L8 be a Remak decomposition of K. 1) The following are equivalent: a) This Remak decomposition of K is essentially unique. b) Every homomorphism αÀ L3 Ä ^ÐKÑ satisfies αÀ L3 Ä ^ÐL3 Ñ. c) There are no nonzero homomorphisms -À L3 Ä ^ÐL4 Ñ for 3 Á 4. 2) If K is either perfect or centerless, then K has an essentially unique Remak decomposition. Proof. For part 2), if K œ K w , then L3 œ L3w for all 3 and so if -À L3 Ä ^ÐL4 Ñ for 4 Á 3, then for +ß , − L3 , -ÐÒ+ß ,ÓÑ œ Ò-+ß -,Ó œ " and so -lL3 œ !. If K is centerless, the result is immediate.…

Exercises 1. 2.

Give an example of an infinite group with a composition series. Find isomorphic refinements of the two series Ö!× – :™ – ™ and Ö!× – ; ™ – ™

3.

where : and ; are distinct primes. Prove that if K and L are groups and if ZÀ Ö"× œ K! – K" – â – K8 œ K and [À Ö"× œ L! – L" – â – L7 œ L are composition series for K and L , respectively, then the series ÐÖ"× { Ö"×Ñ – ÐK" { Ö"×Ñ – â – ÐK8 { Ö"×Ñ – ÐK8 { L" Ñ – â – ÐK8 { L7 Ñ

4. 5. 6. 7.

is a composition series for K { L . How many composition series does a cyclic group of order :8 have? Prove that the multiset of composition factors of a group is an invariant under isomorphism, but not a complete invariant. Prove the uniqueness part of the fundamental theorem of arithmetic using the Jordan–Hölder Theorem. Let K œ G:8 be the direct product of 8 cyclic groups of prime order :. How many compositions series does K have? Hint: K is a vector space.

Finiteness Conditions 8.

289

Prove that a subgroup of a group with a composition series need not have a composition series as follows. Let M œ Ö"ß #ß á × and let K œ WÐMÑ be the restricted symmetric group on M . (Recall from an earlier exercise that this is the set of all permutations of M that have finite support.) For each 8   ", let K8 œ Ö5 − K ± 5B œ B for B  8× and let L8 œ Ö5 − K8 ± 5 lÖ"ßáß8× is even×

9.

a) Show that K œ -8" K8 . b) Show that L œ -8" L8 has index # in K and so L ü K. c) Show that L is simple. d) Show that L contains an infinite abelian subgroup E. e) Show that E has no composition series but that L does. Let c be an isomorphic-invariant property of finite groups. Let K be a group that has c and for which if R ü K, then KÎR has c . Prove that the following are equivalent: a) K has a normal series Ö"× œ K! ü K" ü â ü K8 œ K

whose factor groups have c . b) K has the property that any nontrivial quotient group of K has a nontrivial normal subgroup that has c . Such a group K is said to be hyper-c . 10. Prove the following facts: a) For any +ß , − K, Ò+,ÓL3 œ Ò+ÓL3 Ò,ÓL3 and so the projection map 35 À K Ä L5 is a homomorphism (and an endomorphism of K). b) For any + − K, + œ Ò+ÓL" âÒ+ÓL= c)

The projection map commutes with any inner automorphism #1 of K and therefore preserves normality. 11. Let K have BCC on normal subgroups and suppose that L is a group for which K } K ¸ L } L . Show that K ¸ L .

Chapter 11

Solvable and Nilpotent Groups

Classes of Groups By a class ^ of groups, we mean a subclass of the class of all groups with the following two properties: 1) ^ contains a trivial group 2) ^ is closed under isomorphism, that is, K−^

and

L ¸K

Ê

L −^

For example, the abelian groups form a class of groups. A group of class ^ is called a ^-group and ^-group L that is a subgroup of a group K is called a ^subgroup of K. A class ^ is a trivial class if it contains only one-element groups.

Closure Properties We will be interested in the following closure properties for a class ^ of groups: 1) (Subgroup) K − ^ß

L ŸK

Ê

L−^

2) (Intersection and Cointersection) For Lß O Ÿ K , Lß O − ^ K K ß −^ L O

Ê Ê

L ∩O −^ K −^ L ∩O

3) (Quotient and Extension) For R ü K, K−^ R ß KÎR − ^

Ê Ê

KÎR − ^ K−^

S. Roman, Fundamentals of Group Theory: An Advanced Approach, DOI 10.1007/978-0-8176-8301-6_11, © Springer Science+Business Media, LLC 2012

291

292

Fundamentals of Group Theory

4) (Seminormal Join, Normal Join and Cojoin) For Lß O Ÿ K, Lß O − ^ß one normal in K Lß O − ^ß both normal in K K K ß −^ L O

Ê Ê Ê

LO − ^ LO − ^ K −^ LO

5) (Direct product) Lß O − ^

Ê

L }O − ^

These properties are not independent. Theorem 11.1 The following implications hold for a class ^ of groups: 1) subgroup Ê intersection 2) quotient Ê cojoin 3) seminormal join Ê normal join Ê direct product 4) subgroup and direct product Ê cointersection Thus, a class that is closed under subgroup, quotient, seminormal join, extension is closed under all nine properties above. Proof. Part 1) is clear. For part 2), we have K K LO ¸ ‚ −^ LO L L For part 3), the direct product L { O is the seminormal join of L { Ö"× and Ö"× { O , each of which is in ^ if Lß O − ^. For part 4), if KÎLß KÎO − ^, then K K K ä } −^ L ∩O L O via the map 5À 1ÐL ∩ OÑ È Ð1Lß 1OÑ.… The following definition will prove very convenient. Definition Let ^ be a class of groups. 1) A ^-series is a series whose factor groups belong to the class ^. 2) A ^= -group is a group that has a ^-series and a ^8 -group is a group that has a normal ^-series. 3) The ^= -class is the class of all ^= -groups and the ^8 -class is the class of all ^8 -groups.… Our main interest is in the ^= and ^8 classes in which ^ is either the class of cyclic groups or the class of abelian groups. However, we are also interested in a class of groups that is not a ^= or ^8 class, namely, the nilpotent groups.

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Definition 1) a) A cyclic series is a series that has cyclic factor groups. b) An abelian series is a series that has abelian factor groups. c) A central series for a group K is a normal series Ö"× œ K! ü K" ü â ü K8 œ K for which K5" ÎK5 Ÿ ^ÐKÎK5 Ñ for all 5 . 2) As shown in the table below, we have the following definitions.

Abelian Cyclic Central a) b) c) d)

Series Solvable Polycyclic

Normal Series œ Solvable Supersolvable Nilpotent

A group is solvable if it has an abelian series. A group is polycyclic if it has a cyclic series. A group is supersolvable if it has a normal cyclic series. A group K is nilpotent if it has a central series.…

Note that in previous chapters (see Theorem 7.10), we have found it convenient to use the term central series even when the series does not start at the trivial group, which requires that we distinguish carefully between series in K and series for K. As the title of this chapter suggests, our primary interest is in nilpotent and solvable groups. It is clear that nilpotent groups are solvable. Also, all abelian groups are nilpotent and Theorem 7.10 shows that all finite :-groups are nilpotent: finite : -group or abelian

Ê nilpotent Ê solvable

Note, however, that W$ is solvable but not nilpotent. Also, the symmetric groups W8 are solvable if and only if 8  &. In fact, if 8   &, then W8 has only one nontrivial proper normal subgroup E8 , which is not abelian.

Operations on Series In order to study the closure properties of series-based classes of groups and of the nilpotent class, we must consider various operations on series. Indeed, the operations of intersection, normal lifting, quotient and unquotient as defined in

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Theorem 4.10 can be applied to each step in a series. Specifically, we have the following operations on series. Let Z À K! ü â ü K 8 be a series in K. 1) The intersection of Z with L Ÿ K is Z ∩ L À K! ∩ L ü â ü K8 ∩ L œ L 2) The normal lifting of Z by R ü K is Z R À K! R ü â ü K8 R 3) The quotient of Z by R ü K is Z R K! R K8 R À üâü R R R 4) The unquotient of the series ZÀ

K! K8 üâü R R

where R Ÿ K! is Z Å R À K! ü â ü K8 5) For L3 Ÿ K, the concatenation of the series [ À L! ü â ü L7 and ^ À L7 ü â ü L78 is the series [ ‡ ^ À L! ü â ü L7 ü â ü L78 6) For the series [À L! ü â ü L 8 and Z À K! ü â ü K7 in K, the interleaved series [ µ Z is formed as follows. First, if = œ maxÖ8ß 7×, then we extend whichever series is shorter by repeating the upper endpoint (L8 or K7 ) an appropriate number of times to make the

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295

resulting series of equal length =. Then [ µ Z À ÐL! { K! Ñ ü ÐL" { K! Ñ ü ÐL" { K" Ñ ü ÐL# { K" Ñ ü ÐL# { K# Ñ ü â ü ÐL= { K= Ñ Note that the intersection, normal lifting, quotient, unquotient and interleave of normal series is normal. However, the concatenation of two normal series need not be normal. These operations are used as follows. Theorem 11.2 Let K be a group and let L Ÿ K and R ü K. Let a) Z be a series for K b) [ be a series for L c) a be a series for R d) d be a series for KÎR . Then 1) (Subgroup) Z ∩ L is a series for L 2) (Seminormal join) a ‡ [R is a series for LR 3) (Quotient) ZR ÎR is a series for KÎR 4) (Extension) a ‡ Ðd Å R Ñ is a series for K 5) (Direct product) If Z3 is a series for K3 for 3 œ "ß #, then Z" µ Z# is a series for K" { K# .…

Closure Properties of Groups Defined by Series Theorem 4.10 and the previous theorems provide the following facts about closure of ^= -classes and ^8 -classes. Theorem 11.3 Let ^ be a class of groups. 1) (Subgroup) If ^ is closed under subgroup, then the ^= and ^8 classes are closed under subgroup. 2) (Seminormal join) If ^ is closed under quotient, then the ^= -class is closed under seminormal join. 3) (Quotient) If ^ is closed under quotient, then the ^= and ^8 classes are closed under quotient. 4) (Extension) The ^= -class is closed under extension. 5) (Direct product) The ^= and ^8 classes are closed under direct product. In particular, if ^ is closed under subgroup and quotient, then the ^= -class is closed under the following eight operations: 6) subgroup 7) quotient 8) intersection 9) cointersection 10) cojoin 11) direct product

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12) seminormal join 13) extension and the ^8 -class is closed under all of these operations except seminormal join and extension. Proof. For 1), if ^ is closed under subgroup, then (normal) ^-series are closed under intersection and so the ^= and ^8 classes are closed under subgroup. For 2) and 3), if ^ is closed under quotient, then ^-series are closed under normal lifting. Hence, a ‡ [R and Z R ÎR are ^-series. It follows that the ^= -class is closed under seminormal join and the ^= and ^8 classes are closed under quotient. For 5), if Z3 is a (normal) ^-series for K3 , then Z" µ Z# is a (normal) ^-series for K" { K# .… Thus, since the classes of cyclic groups and abelian groups are closed under subgroup and quotient, we have the following. Theorem 11.4 The polycyclic, solvable and supersolvable classes are closed under the following eight operations (except where noted): 1) subgroup 2) quotient 3) intersection 4) cointersection 5) cojoin 6) direct product 7) seminormal join (except for supersolvable) 8) extension (except for supersolvable).… Let us now turn to the closure properties of nilpotent groups. Theorem 11.5 1) Central series are closed under intersection, normal lifting, quotient and unquotient. 2) The class of nilpotent groups has the following seven closure properties: a) subgroup b) quotient c) normal join (this is Fitting's theorem, to be proved a bit later) d) direct product e) intersection f) cointersection g) cojoin but not extension. Proof. Part 1) follows from Theorem 4.10 and Theorem 4.11. The statement about extensions in part 3) can be verified by looking at W$ .…

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297

Nilpotent Groups We now undertake a closer look at nilpotent groups. We will prove that a finite group K is nilpotent if and only if it is the direct product of :-groups and so Theorem 8.11 shows that finite nilpotent groups have many other interesting characterizations.

Higher Centers An extension L ü O in K is central in K if and only if O K Ÿ ^Œ L L and so the largest subgroup O of K for which L ü O is central in K is the subgroup O for which O K œ ^Œ L L With this in mind, we define a function ' œ 'K on norÐKÑ by ^Œ

K R

œ

'K ÐR Ñ R

for R ü K. Note also that 'K ÐR Ñ K œ ^Œ R R

«

K R

and so 'K ÐR Ñ « K. Writing ' 5 ÐÖ"×Ñ as ' 5 Ð"Ñ, the proper series ' ! Ð"Ñ ö ' " Ð"Ñ ö ' # Ð"Ñ ö â is called the upper central series for K and each ' 5 Ð"Ñ is called a higher center of K. The first higher center is the center ^ÐKÑ of K . To see that the upper central series ascends more rapidly than any other central series of the form Ö"× œ K! ü K" ü K# ü â we have 'ÐK5 Ñ   K5" for all 5 and it is clear that K5 Ÿ ' 5 Ð"Ñ holds for 5 œ !. If this holds for a particular value of 5 , then the monotonicity of ' implies that

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K5" Ÿ ' ÐK5 Ñ Ÿ ' Ð' 5 Ð"ÑÑ œ ' 5" Ð"Ñ Hence, K5 Ÿ ' 5 Ð"Ñ for all 5 . Theorem 11.6 Let K be a nilpotent group. The upper central series ' ! Ð"Ñ ö ' " Ð"Ñ ö ' # Ð"Ñ ö â for K is characteristic and ascends more rapidly than any other central series for K, that is, if Ö"× œ K! ü K" ü K# ü â is central in K, then K5 Ÿ ' 5 Ð"Ñ for all 5   !. 1) K is nilpotent if and only if the upper central series reaches K. 2) If K is nilpotent, then all central series for K have length greater than or equal to the length of the upper central series for K.… The higher centers have some important applications. Theorem 11.7 Let K be nilpotent, with higher centers ' 5 Ð"Ñ. 1) If L Ÿ K, then L ' 5 Ð"Ñ ü L ' 5" Ð"Ñ 2) If R ü K, then R ∩ ' 5 Ð"Ñ œ Ö"×

Ê

R ∩ ' 5" Ð"Ñ Ÿ ^ÐKÑ

As a consequence, 3) K has the property that every subgroup is subnormal 4) K has the center-intersection property 5) Every chief series for K is central. Proof. For part 1), since ÒKß ' 5" Ð"ÑÓ Ÿ ' 5 Ð"Ñ, Theorem 3.41 implies that ÒL ' 5 Ð"Ñß L ' 5" Ð"ÑÓ œ ÒL ' 5 Ð"Ñß LÓÒL ' 5 Ð"Ñß ' 5" Ð"ÑÓL But each factor on the right is contained in L ' 5 Ð"Ñ and so L ' 5 Ð"Ñ ü L ' 5" Ð"Ñ. Part 3) follows from part 1), since we may lift the upper central series for K by L to get a series from L to K, whence L is subnormal in K. For part 2), since Ò' 5" Ð"Ñß KÓ Ÿ ' 5 Ð"Ñ and ÒR ß KÓ Ÿ R , it follows that ÒR ∩ ' 5" Ð"Ñß KÓ Ÿ R ∩ ' 5 Ð"Ñ œ Ö"× and so R ∩ ' 5" Ð"Ñ Ÿ ^ÐKÑ. For part 4), there is a largest 5 for which

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299

R ∩ ' 5 Ð"Ñ œ Ö"× and so R ∩ ' 5" Ð"Ñ is a nontrivial subgroup of ^ÐKÑ. For part 5), the factor group K5" ÎK5 of a chief series Z for K is a minimal normal subgroup of the nilpotent group KÎK5 and so the center-intersection property implies that K5" ÎK5 is central. Thus, Z is central.… We can now augment Theorem 8.11 by adding the nilpotent condition. Theorem 11.8 The following are equivalent for a finite group K: 1) K is nilpotent. 2) Every Sylow subgroup of K is normal. 3) K is the direct product of its Sylow :-subgroups K œ  ]: :−c

4) If L Ÿ K, then L œ  ÐL ∩ ]: Ñ :−c

5) (Strong converse of Lagrange's theorem) If 8 ± 9ÐKÑ, then K has a normal subgroup of order 8. 6) K is the direct product of :-subgroups. 7) Every subgroup of K is subnormal. 8) K has the normalizer condition. 9) Every maximal subgroup of K is normal. 10) KÎFÐKÑ is abelian. Proof. Theorem 8.11 states that 2)–10) are equivalent. Moreover, Theorem 7.10 implies that a finite :-group is nilpotent and therefore so is any direct product of finite :-groups. Hence, 6) implies 1). Theorem 11.7 shows that 1) implies 7).…

Lower Centers In terms of commutators, an extension L ü O in K is central in K if and only if ÒOß KÓ Ÿ L and so L œ ÒOß KÓ is the smallest subgroup O of K for which L ü O is central in K; in fact, ÒOß KÓ « O . Thus, if we define the “commutator with K” function > œ >K on subÐKÑ by >K ÐOÑ œ ÒOß KÓ then >K ÐOÑ is the smallest subgroup of K for which >K ÐOÑ Ÿ O is central in K. The (possibly infinite) descending central series â ö ># ÐKÑ ö >" ÐKÑ ö >! ÐKÑ œ K is called the lower central series for K and each >5 ÐKÑ is called a lower center of K. The first lower center is the commutator subgroup >ÐKÑ œ Kw .

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Fundamentals of Group Theory

An induction argument shows that the lower central series descends more rapidly than any other central series, in the sense that if â ü K# ü K" ü K! œ K is central in K, then >5 ÐKÑ Ÿ K5 for all 5   !. For if >5 ÐKÑ Ÿ K5 , then the monotonicity of > implies that >5" ÐKÑ œ >Ð>5 KÑ Ÿ >ÐK5 Ñ Ÿ K5" Theorem 11.9 Let K be a group. The lower central series â ö ># ÐKÑ ö >" ÐKÑ ö >! ÐKÑ œ K for K is characteristic and descends more rapidly than any central series for K, that is, if â ü K# ü K" ü K! œ K is central in K, then >5 ÐKÑ Ÿ K5 for all 5   !. 1) K is nilpotent if and only if the lower central series reaches Ö"×. 2) If K is nilpotent, then all central series for K have length greater than or equal to the length of the lower central series for K.… The commutator function >K has some simple but useful properties that are consequences of Theorems 3.40 and 3.41. Theorem 11.10 Let K be a group and let L ß O ß P ü K. 1) >L ÐOÑ ü K 2) >L ÐOÑ œ >O ÐLÑ 3) >L ÐOPÑ œ >L ÐOÑ>L ÐPÑ and >LO ÐPÑ œ >L ÐPÑ>O ÐPÑ 4) >L is deflationary, that is, >L ÐOÑ Ÿ L In fact, >L ÐOÑ Ÿ L ∩ O Hence, for all 5   ", 5 >5L ÐLÑ Ÿ >K ÐKÑ

Solvable and Nilpotent Groups

301

5) If R ü K and R Ÿ L ∩ O , then for any 5   ", >5OÎR Œ

L R

6) For 5   ",

œ

>5O ÐLÑR R

$

>5LO ÐLOÑ œ

>E" â>E5 ÐFÑ

E" ßáßE5 ßF−ÖLßO×

Proof. Part 5) holds for 5 œ " since >OÎR Œ

L R

œ”

L O ÒLß OÓR >O ÐLÑR ß •œ œ R R R R

and if it holds for a particular value of 5 , then >5" OÎR Œ

L R

œ ”>5OÎR Œ

L O ß • R R >5O ÐLÑR O œ” ß • R R Ò>5O ÐLÑß OÓR œ R >5" O ÐLÑR œ R

and so this holds for all 5   ". For part 6), we have >LO ÐLOÑ œ >L ÐLÑ>O ÐLÑ>L ÐOÑ>O ÐOÑ œ

$

>E ÐFÑ

EßF−ÖLßO×

and an easy induction proves the general formula.…

Nilpotency Class Theorem 11.9 and Theorem 11.6 imply that for a nilpotent group, the upper and lower central series have the same length. Definition Let K be a nilpotent group. The common length of the upper and lower central series is called the nilpotency class of K, which we denote by nilclassÐKÑ.… Moreover, if K is nilpotent and Z À Ö"× œ K! – â – K7 œ K is a central series for K of length 7, then

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Fundamentals of Group Theory

5 >75 K ÐKÑ Ÿ K5 Ÿ 'K Ð"Ñ

for all 5 œ !ß á ß 7, where >3 ÐKÑ œ Ö"× and ' 3 Ð"Ñ œ K if 3   nilclassÐKÑ. The nilpotent groups of class ! are the trivial groups and the nilpotent groups of class " have ^ÐKÑ œ K or equivalently, Kw œ Ö"× and so are the nontrivial abelian groups. A group K has nilpotency class # if and only if either of the following conditions holds: 1) Ö"×  Kw  K and ÒKw ß KÓ œ Ö"×, or equivalently, Ö"×  Kw Ÿ ^ÐKÑ  K 2) Ö"×  ^ÐKÑ  K and ^Œ

K ^ÐKÑ

œ

K ^ÐKÑ

or equivalently, K is not abelian but KÎ^ÐKÑ is abelian. Theorem 3.42 implies that Kw Ÿ ^ÐKÑ if and only if ÒÒ+ß ,Óß -Ó œ Ò+ß Ò,ß -ÓÓ for all +ß ,ß - − K . Hence, for nonabelian groups, this condition is equivalent to being nilpotent of class #. Theorem 11.11 Let K be nilpotent. 1) If L Ÿ K, then nilclassÐLÑ Ÿ nilclassÐKÑ 2) If R ü K, then nilclassÐKÎR Ñ Ÿ nilclassÐKÑ 3) (Fitting's theorem) The join of two normal nilpotent subgroups of a group K is nilpotent. In fact, if Lß O ü K , then nilclassÐLOÑ Ÿ nilclassÐLÑ  nilclassÐOÑ Proof. The first two parts follow from Theorem 11.10. For part 3), Theorem 11.10 implies that >5LO ÐLOÑ œ

$

>E" â>E5 ÐFÑ

E" ßáßE5 ßF−ÖLßO×

for all 5   ". Now, suppose that nilclassÐLÑ œ - and nilclassÐOÑ œ . and let 5 œ -  .. Then each factor on the right above has the form

Solvable and Nilpotent Groups

303

P œ >E" â>E-. ÐFÑ Among the subgroups E" ß á ß E-. ß F , suppose there are 2 L 's and 5 O 's, where 2  5 œ -  .  ". Then either 2   -  " or 5   .  " and we may assume without loss of generality that 2   -  ". Since >O is deflationary, removing all >O 's from the expression for P results in a possibly larger subgroup Q œœ

>2L ÐOÑ >2" L ÐLÑ

if F œ O if F œ L

However, in the former case, 2" 2" >2L ÐOÑ œ >2" L >L ÐOÑ œ >L >O ÐLÑ Ÿ >L ÐLÑ

and so P Ÿ Q Ÿ >2" L ÐLÑ Ÿ >L ÐLÑ œ Ö"×

Hence, >-. LO ÐLOÑ œ Ö"× which implies that LO is nilpotent of class at most -  . .…

An Example We now describe a family of groups showing that for any -   !, there are nilpotent groups of nilpotency class - . Let `8 ÐVÑ be the family of all 8 ‚ 8 matrices over a commutative ring V with identity and let Y œ Y X Ð8ß VÑ be the unitriangular matrices over V . (Recall that a matrix is unitriangular if it is upper triangular, with "'s on the main diagonal.) Denote the Ð3ß 4Ñth entry in Q by Q3ß4 . For 5   !, the 5 th superdiagonal Q Ð5Ñ of a matrix Q are the elements of the form Q3ß35 . For ! Ÿ 5 Ÿ 8  ", let R5 be the set of all 8 ‚ 8 matrices over V with !'s on or below the 5 th superdiagonal, that is, for E − `8 ÐVÑ, E − R5

Í

E3ß4 œ ! for all 4 Ÿ 3  5

It is routine to confirm that R5 R7 © R57" In particular, R5 R5 © R5 For 5   !, let Y5 œ ÖM×  R5 œ ÖM  E ± E − R5 ×

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Fundamentals of Group Theory

To see that Y5 is a subgroup of Y , we have Y5 Y5 œ ÐÖM×  R5 ÑÐÖM×  R5 Ñ © ÖM×  R5  R5  R5 R5 © Y5 and if E − R5 , then since E? œ ! for some ?   !, it follows that ÐM  EÑ" œ M  E  E#  â „ E?" − Y5 As to commutators, if E − R5 and F − R7 , then ÒM  Eß M  FÓ œ ÐM  EÑÐM  FÑÐÐM  FÑÐM  EÑÑ" œ ÐM  E  F  EFÑÐM  F  E  FEÑ" œ ÐM  \ÑÐM  ] Ñ" œ M  Ð\  ] ÑÐM  ] Ñ" where \ œ E  F  EF

and ] œ F  E  FE

Moreover, \  ] œ EF  FE − R57" implies that ÒM  Eß M  FÓ − Y57" and so ÒY5 ß Y7 Ó Ÿ Y57" Taking 7 œ ! gives ÒY5 ß Y Ó Ÿ Y5" Ÿ Y5 which shows both that Y5 is normal in Y and that Y5" – Y5 is central in Y . Thus, the series ÖM× œ Y8" – â – Y" – Y! œ Y is central and so Y is nilpotent of class at most 8  ". On the other hand, let I3ß4 denote the matrix with all !s except for a " in the Ð3ß 4Ñthe position. Then for 3 Á 4  ", an easy calculation shows that ÒM  I3ß4 ß M  I4ß4" Ó œ M  I3ß4" In particular, if 8   $, then ÒM  I"ß# ß M  I#ß$ Ó œ M  I"ß$ and so >ÐKÑ œ ÒKß KÓ Á ÖM×. Also, if 8   %, then ÒM  I"ß$ ß M  I$ß% Ó œ M  I"ß% and so ># ÐKÑ œ Ò>ÐKÑß KÓ Á ÖM×. More generally,

Solvable and Nilpotent Groups

305

ÒM  I"ß8" ß M  I8"ß8 Ó œ M  I"ß8 Á M and so >8# ÐKÑ Á ÖM×, which shows that Y is nilpotent of class 8  ".

Solvability We now turn to a discussion of solvable groups.

Perspective on Solvability Solvable groups have played an extremely important role in the study of the location of the roots of polynomials over a field J . Let us pause to describe this role in general terms. For more details, we refer to reader to Roman, Field Theory [27]. If J is a subfield of a field I , we say that J Ÿ I is a field extension. Associated to each field extension J Ÿ I is a group KJ ÐIÑ, called the Galois group of the extension and defined as the group of all (ring) automorphisms 5 of I that fix the elements of J , that is, for which 5+ œ + for all + − J . It turns out that the properties of the “simpler” Galois group can often shed considerable light on properties of the “more complex” field extension. To illustrate, one of the principal motivations for the development of abstract algebra since, oh say 3000 B.C., has been the desire (expressed in one form or another) to find the roots of a polynomial :ÐBÑ with coefficients from a given base field J . In fact, we now know that for any nonconstant polynomial :ÐBÑ of degree ., there is an extension J of J , called an algebraic closure of J , that contains a full set of . roots for :ÐBÑ. Moreover, lying between the fields J and J is the smallest field I containing these roots of :ÐBÑ, called a splitting field for :ÐBÑ. The desire to express the roots of :ÐBÑ by arithmetic formula (similar to the quadratic formula) or to show that this could not be done is what motivated Galois to first define some version of what we now know as a group. The idea of expressing the roots of a polynomial :ÐBÑ by formula means that, starting with the elements of the base field J , we can “capture” all of the roots of :ÐBÑ through a finite number of special types of extensions of J . In particular, for each extension, we are allowed to include an 8th root of an existing element and only whatever else is required in order to make a field. Specifically, for the first extension, we may choose any ?! − J and any root 8!
8! where J ÐÈ ?! Ñ is the smallest subfield of J containing J and
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Fundamentals of Group Theory

extensions produce a tower of fields of the form

85 8" 8! 8! 8! J Ÿ J ÐÈ ?! Ñ Ÿ J Ð È ?! ß È ?" Ñ Ÿ â Ÿ J Ð È ?! ß á ß È ?5 Ñ

where each ?3 is an element of the immediately preceeding field. This type of tower of fields is called a radical series. If all of the roots of a polynomial :ÐBÑ lie within a radical series over J , then we say that the polynomial equation :ÐBÑ œ ! is solvable by radicals. (For simplicity, we assume that J has characteristic !.) It is not hard to show that a polynomial equation :ÐBÑ œ ! is solvable by radicals if and only if the roots of :ÐBÑ can be captured within a finite tower of fields J Ÿ J Ðα" Ñ Ÿ J Ðα" ß α# Ñ Ÿ â Ÿ J Ðα" ß á ß α7 Ñ where each field has prime degree over the previous field, that is, the dimension of each field as a vector space over the previous field is a prime number. Now let I be the splittting field for :ÐBÑ in J . If the radical series above does capture the roots of :ÐBÑ, that is, if I Ÿ J Ðα" ß á ß α7 Ñ, then taking Galois groups (which reverses inclusion) gives the descending sequence ZÀ KJ ÐIÑ   KJ Ðα" Ñ ÐIÑ   â   KJ Ðα" ßáßα7 Ñ ÐIÑ But KJ Ðα" ßáßα7 Ñ ÐIÑ  KI ÐIÑ œ Ö+× and so the sequence Z reaches the trivial group. Galois showed that (in modern terminology) Z is an abelian series and so KJ ÐIÑ is solvable. Thus, Galois proved that if :ÐBÑ œ ! is solvable by radicals, then its Galois group KJ ÐIÑ is solvable. He also proved the converse. Now, an element 5 − KJ ÐIÑ of the Galois group fixes the coefficients of :ÐBÑ, since they lie in J . Hence, 5 must send a root < of :ÐBÑ in I to another root of :ÐBÑ in I , that is, the elements of KJ ÐIÑ are permutations of the set of roots of :ÐBÑ in I . Moreover, since I is generated by these roots, each 5 − KJ ÐIÑ is uniquely determined by its behavior on these roots. Thus, one often thinks of KJ ÐIÑ simply as a subgroup of the permutation group W8 , where 8 œ degÐ:ÐBÑÑ. In fact, there are many cases in which KJ ÐIÑ, thought of as a subgroup of W8 , is actually W8 itself. For example, it can be shown for any prime :, the Galois group of the splitting field of any irreducible polynomial 0 ÐBÑ − ÒBÓ of degree : with exactly two nonreal roots is W: .

Solvable and Nilpotent Groups

307

However, W8 is not solvable for 8   &, since E8 is simple and so the only nontrivial series for W8 is Ö+× – E8 – W8 , which is not abelian. Thus, the roots of the polynomials described above cannot be captured within a radical series, that is, these polynomials are not solvable by radicals. Note that this shows that there are individual polynomials that are not solvable by radicals. Thus, not only is there no general formula, similar to the quadratic, cubic and quartic formulas, for the solutions of arbitrary quintic equations, but there are even individual quintic equations whose solutions are not obtainable by formula! Galois used his remarkable theory in his paper Memoir on the Conditions for Solvability of Equations by Radicals of 1831 (but not published until 1846!), to show that the general equation of degree & or larger is not solvable by radicals. (Proofs that the &th degree equation is not solvable by radicals were offered earlier: An incomplete proof by Ruffini in 1799 and a complete proof by Abel in 1826.) Thus, the notion of solvability arose through the desire to settle the question of whether we could solve all polynomial equations by simple formula. Of course, solvable groups are important for other reasons. In fact, we will see that the class of solvable groups has a sort of super-Sylow theorem, to wit, if K is solvable of order 78 where Ð7ß 8Ñ œ ", then K has a (Hall) subgroup of order 7 and all subgroups of order 7 are conjugate.

The Derived Series For any solvable group, there is an abelian series that descends more rapidly than any other abelian series. Moreover, this series is also characteristic in K. A series Ö"× œ K! ü K" ü â ü K7 œ K is abelian if and only if w K5" Ÿ K5

for all 5 œ !ß á ß 7  ", where the prime denotes commutator subgroup. Definition Let K be a group. The subgroups defined by KÐ!Ñ œ K,

KÐ"Ñ œ Kw

and, in general for 8   ", KÐ8Ñ œ ÐKÐ8"Ñ Ñw are called the higher commutators of K. The group KÐ8Ñ is called the 8th commutator subgroup of K. The series of higher commutators:

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â ö KÐ$Ñ ö KÐ#Ñ ö KÐ"Ñ ö K is called the derived series for K.… The monotonicity of the commutator operation implies that the derived series descends from K more rapidly than any other abelian series. Moreover, since KÐ5"Ñ « KÐ5Ñ , the derived series is characteristic. Theorem 11.12 Let K be a group. 1) The derived series â ö KÐ$Ñ ö KÐ#Ñ ö KÐ"Ñ ö K is the abelian series of steepest descent, in the sense that if the series âK$ ü K# ü K" ü K! œ K is abelian, then KÐ5Ñ Ÿ K5 for all 5 . 2) K is solvable if and only if its derived series reaches the trivial group, that is, if and only if there is a 5   " for which KÐ5Ñ œ Ö"×. The smallest integer 8 for which KÐ8Ñ œ Ö"× is called the derived length of K, which we denote by derlenÐKÑ. 3) A group K is solvable if and only if it has a normal abelian series. 4) The length of any abelian series for K is greater than or equal to the derived length of K.… We will have use for the following fact about higher commutators of quotient groups. Theorem 11.13 Let K be a group and let R ü K. Then K Œ R

Ð8Ñ

œ

K Ð8Ñ R R

for all 8   !. Proof. For any R Ÿ L ü K, we have Œ

L R

w

œ”

L L ÒLß LÓR L wR ß •œ œ R R R R

In particular, for L œ K, we have Œ

K R

w

œ”

K K ÒKß KÓR Kw R ß •œ œ R R R R

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309

and so the result holds for 8 œ ". Assuming that the result holds for an arbitrary 5 , we have with L œ KÐ8Ñ R , K Œ R

Ð8"Ñ

K œ –Œ R

Ð8Ñ w

— œ

KÐ8Ñ R R

w

œ

ÐKÐ8Ñ R Ñw R R

Finally, Theorem 3.41 implies that ÐKÐ8Ñ R Ñw R œ ÒKÐ8Ñ R ß K Ð8Ñ R ÓR œ ÒK Ð8Ñ ß K Ð8Ñ ÓR œ K Ð8"ÑR and so the result follows.… Theorem 11.14 If K is solvable, Eß Fß L Ÿ K and R ß O ü K, then 1) derlenÐLÑ Ÿ derlenÐKÑ 2) derlenÐKÎR Ñ Ÿ derlenÐKÑ 3) derlenÐKÑ Ÿ derlenÐR Ñ  derlenÐKÎR Ñ 4) derlenÐE  FÑ Ÿ maxÖderlenÐEÑß derlenÐFÑ×. Proof. For part 1), since L Ð5Ñ Ÿ KÐ5Ñ for all 5   !, if 8 œ derlenÐKÑ then L Ð8Ñ Ÿ KÐ8Ñ œ Ö"×. Thus, derlenÐLÑ Ÿ 8. For part 2), Theorem 11.13 implies that if KÐ8Ñ œ Ö"×, then ÐKÎR ÑÐ8Ñ œ ÖR ×. For part 3), if KÎR has derived length ., then KÐ.Ñ R ÎR œ ÖR × and so KÐ.Ñ Ÿ R . If the derived length of R is /, then KÐ./Ñ Ÿ R Ð/Ñ œ Ö"× and so the derived length of K is at most .  /. Part 4) follows from the fact that ÐE  FÑw œ Ew  F w .…

Properties of Solvable Groups If R is a minimal normal subgroup of a group K and if Z À Ö"× œ K. ü â ü K! œ K is any normal series in K , then R Ÿ K3 or else R ∩ K3 œ Ö"× for all 3 and so there is an index 5 for which R Ÿ K5

and R ∩ K5" œ Ö"×

Therefore, if K is solvable and if Z is the derived series for K , then R Ÿ KÐ5Ñ

and

R ∩ KÐ5"Ñ œ Ö"×

and so R w Ÿ R ∩ KÐ5"Ñ œ Ö"×, whence R is abelian. Theorem 11.15 Let K be solvable. Any minimal normal subgroup R of K is abelian. Moreover, if R contains a nontrivial element of finite order, then R is elementary abelian.

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Proof. For the final statement, R has an element of prime order : and since R: ³ ÖB − R ± B: œ "× ü R it follows that R œ R: is an elementary abelian group.… If K is solvable and has a composition series, then the factor groups of the composition series are both simple and solvable and therefore cyclic of prime order. Theorem 11.16 The following are equivalent for a group K that has a composition series. 1) K is solvable. 2) Every composition series for K has prime order factor groups. 3) K has a cyclic series in which each factor group has prime order. 4) K has a cyclic series, that is, K is polycyclic. 5) Every chief series for K has factor groups that are elementary abelian. Proof. We have seen that 1) implies 2) and it is clear that 2) implies 3), that 3) implies 4) and that 4) implies 1). Thus, 1)–4) are equivalent. It is clear that 5) implies 1). If K is solvable, the factor groups K5" ÎK5 of a chief series are minimal normal in the solvable group KÎK5 and so are elementary abelian by Theorem 11.15.… The following theorem contains some sufficient (but not necessary) conditions for solvability. The proof of the Feit–Thompson Theorem is quite involved, running almost 300 pages. For a proof of the Burnside result, we refer the reader to Robinson [26]. Theorem 11.17 1) (Feit–Thompson Theorem) Any group of odd order is solvable; equivalently, every finite nonabelian simple group has even order. 2) (Burnside :; Theorem) Every group of order :7 ; 8 where : and ; are primes, is solvable. Proof. The equivalence in part 1) is left as an exercise.…

Hall's Theorem on Solvable Groups Let K be a finite group. Recall that a Hall subgroup L of K is a subgroup with the property that its order 9ÐLÑ and index ÐK À LÑ are relatively prime. The Schur–Zassenhaus Theorem tells us that every normal Hall subgroup has a complement and that all such complements are conjugate. As to the existence of Hall subgroups, the Sylow theorems tell us that if 9ÐKÑ œ :5 7 with : prime and Ð:5 ß 7Ñ œ ", then K has a Hall (Sylow) subgroup of order :5 and that all such subgroups are conjugate. In 1928, Philip

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Hall showed that for a finite solvable group, this result applies not just to prime power factors. Theorem 11.18 (Hall's theorem, 1928) Let K be a finite solvable group with 9ÐKÑ œ +, , where Ð+ß ,Ñ œ ". Then K has a Hall subgroup of order + and all subgroups of order + are conjugate. Proof. We may assume that +ß ,  ". The proof is by induction on 9ÐKÑ. If 9ÐKÑ œ ", the result holds trivially. Assume that it holds for all groups of order less than 9ÐKÑ. If K does not have a minimal normal subgroup, then K is simple and solvable and therefore cyclic of prime order and so +ß ,  " is false. Thus, K has a minimal normal subgroup R , which as we have seen, is an elementary abelian group of prime power order :7 . There are cases to consider, based on whether :7 ± + or :7 ± , . Case 1: 9ÐR Ñ œ :7 œ , In this case, R is a normal Hall subgroup of K. Hence, the Schur–Zassenhaus Theorem implies that R has a complement and all such complements are conjugate. But the complements of R are precisely the subgroups of order +. Case 2: 9ÐR Ñ œ :7 ± , and :7  , If :7 ± , but :7 Á , , then 9ÐKÎR Ñ œ +Ð,Î:7 Ñ  +, and so the inductive hypothesis implies that KÎR has a subgroup OÎR of order +. Hence, 9ÐOÑ œ +:7  +, and the inductive hypothesis applied to O shows that O (and hence K) has a subgroup L of order +. As to conjugation, if 9ÐL" Ñ œ 9ÐL# Ñ œ +, then L3 ∩ R œ Ö"× and so 9ÐL3 R ÎR Ñ œ +. Hence, the inductive hypothesis implies that Œ

L# R R

BR

œ

L" R R

for some B − K and so L#B R œ L" R But 9ÐL" R Ñ œ 9ÐKÑ and L" and L#B are Hall subgroups of L" R of order +. Hence, the induction hypothesis implies that L" and L#B are conjugate in L" R , whence in K. Case 3: 9ÐR Ñ œ :7 ± + If 9ÐR Ñ œ :7 ± +, then 9ÐKÎR Ñ œ Ð+Î:7 Ñ,  +, and the inductive hypothesis implies that KÎR has a subgroup OÎR of order +Î:7 , whence 9ÐOÑ œ +. As to conjugacy, if 9ÐLÑ œ +, then R Ÿ L , for if not, then 9ÐLR Ñ ± 9ÐLÑ9ÐR Ñ œ +: 7 and so R L is a subgroup of K of order greater than + but relatively prime to , ,

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which contradicts Lagrange's theorem. Therefore, if 9ÐL" Ñ œ 9ÐL# Ñ œ +, then L" ÎR and L# ÎR are Hall subgroups of KÎR of order +Î:7 and so L" ÎR and L# ÎR are conjugate in KÎR , whence L" and L# are conjugate in K.… A sort of converse of the previous theorem also holds. The proof uses the Burnside :; theorem (Theorem 11.16). Definition Let K be a finite group and let : be a prime for which 8 œ :5 7, where Ð7ß :Ñ œ " and 5   ". Then a Hall :w -subgroup of K is a subgroup L of order 7.… Theorem 11.19 If a finite group K has a Hall :w -subgroup for every prime : dividing 9ÐKÑ, then K is solvable. Proof. Assume that the theorem is false and let K be a counterexample of smallest order. If K is not simple, then let R be a nontrivial proper normal subgroup of K. We leave it as an exercise to show that R ∩ L is a Hall :w subgroup of R and LR ÎR is a Hall :w -subgroup of KÎR . Hence, R and KÎR are solvable and therefore so is K, a contradiction. Hence, K is simple. Now suppose that 9ÐKÑ œ :"/" â:8/8 , where the :3 's are distinct primes and /3   ". The Burnside :; theorem implies that 5   $. If K3 is a Hall :3w -subgroup of K, then 9ÐK3 Ñ œ 9ÐKÑÎ:3/3 and the Poincaré theorem and the fact that the indices ÐK À K3 Ñ are pairwise relatively prime imply that for any 5 of the groups K3 , ÐK À K3" ∩ â ∩ K35 Ñ œ $4 :34 4 /3

and so kK3" ∩ â ∩ K35 k œ

9ÐKÑ / #4 :3 34 4

Also, for any 3, ¹K3 ì ,4Á3 K4 ¹ œ kK3 k¹,4Á3 K4 ¹ œ and so

9ÐKÑ 9ÐKÑ œ 9ÐKÑ :3/3 #4Á3 :4/4

K3 ì ,4Á3 K4 œ K

If L œ K$ ∩ â ∩ K8 , then 9ÐLÑ œ :"/" :#/# , which is solvable by the Burnside :; theorem. If L is simple, then L is abelian and so L is cyclic of prime order, which is false. Hence, let R be a minimal normal subgroup of L . Then R is elementary abelian of exponent, say :" and so is contained in any Sylow :" subgroup of L . But K# ∩ L has order :"/" and so R ü K# ∩ L . Now,

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R K" ∩L Ÿ R and so R K Ÿ R K# ÐK" ∩LÑ Ÿ R K# Ÿ K# and so the normal closure R K is a proper nontrivial normal subgroup of K, a contradiction.…

Exercises 1. 2.

Show that the classes of cyclic groups, abelian groups and nilpotent groups do not have the extension property. Show that if E – F is abelian or cyclic, then any refinement EüLüF is also abelian or cyclic.

Nilpotent Groups 3. 4. 5. 6. 7. 8. 9.

Can a nontrivial centerless group be nilpotent? a) Prove that any finite nilpotent group is supersolvable. b) Find an example to show that not every finite supersolvable group is nilpotent. Let K be a finite group. Prove that K is nilpotent if and only if every nontrivial quotient group of K has a nontrivial center. Let K be nilpotent but not abelian. Let E Ÿ K be maximal with respect to being normal in K and abelian. Prove that E œ GK ÐEÑ. Hint: Show that GÎE ü KÎE and that GK ÐEÑÎE ∩ ^ÐKÎEÑ Á Ö"×. For any even positive integer 8, prove that every group of order 8 is nilpotent if and only if 8 is a power of #. Prove that a nilpotent group is supersolvable if and only if it satisfies the ascending chain condition on subgroups. If L is nilpotent of class - and O is nilpotent of class . , prove that L } O is nilpotent of class maxÐ-ß .Ñ.

Solvable Groups 10. Prove that W% is solvable but not supersolvable. 11. Prove that W8 is solvable for 8 Ÿ %. 12. Assuming that E& is the smallest nonabelian simple group (which it is), prove that every group of order less than '! is solvable. 13. Let R be a nontrivial proper normal subgroup of a group K. Let : be a prime and : ± 9ÐR Ñ. Let L be a Hall :w -subgroup of K. Prove that R ∩ L and R LÎR are Hall :w -subgroups of R and KÎR , respectively. 14. Prove that the following are equivalent for a finite group K: a) K is solvable. b) Every nontrivial normal subgroup of K has a nontrivial abelian quotient group. c) Every nontrivial quotient group of K has a nontrivial abelian normal subgroup.

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Fundamentals of Group Theory

/7 15. Let K be a finite group of order 8 œ :"/" â:7 . Prove that K is solvable if and only if the composition length of K is - œ /3 . 16. Prove that a solvable group with a composition series must be finite. 17. Let K be a nontrivial finite solvable group. a) Prove that b: ÐKÑ is nontrivial for some prime :, that is, K has a nontrivial normal :-subgroup. b) Prove that b; ÐKÑ is nontrivial for some prime ;, that is, K has a normal subgroup L for which KÎL is a nontrivial ; -group. 18. Let K be a finite group. Prove directly that an abelian series can always be refined into a cyclic series with prime order factor groups. 19. Prove that the following are equivalent: a) Any finite group of odd order is solvable. b) Any finite nonabelian simple group has even order. 20. a) Prove that a finite group K is solvable if and only if W w Á W for all subgroups W Á Ö"× of K. b) Prove that if K contains a nonabelian simple subgroup W , then K is not solvable. c) Show that W w Á W for all subgroups of the dihedral group H#8 , showing that H#8 is solvable. Hint: Find an abelian subgroup R of index #. How do subgroups interact with R ? 21. A subgroup L of a group K is abnormal if

+ − ØLß L + Ù for all + − K. Prove that the normalizer of a Hall subgroup of a solvable group is abnormal.

Polycyclic Groups 22. a)

Let K be a polycyclic group with a cyclic series of length 8. Prove that K is 8-generated. b) Let E be an 8-generated abelian group for 8   ". Prove that E is polycyclic. 23. Prove that the following are equivalent: a) K is polycyclic. b) Every subgroup of K is finitely generated and solvable. c) Every normal subgroup of K is finitely generated and solvable. 24. Prove that a group K is polycyclic if and only if it is solvable and satisfies the maximal condition on subgroups, that is, if and only if every nonempty collection of subgroups of K as a maximal member. 25. a) Let E  F have an infinite cyclic factor group. Let E œ L!  L"  â  L 7 œ F be a proper refinement of E  F . Describe the factor groups of this refinement.

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315

b) Let K be polycyclic. Prove that the number of steps whose factor group is infinite is the same for all cyclic series for K. Hint: Any two cyclic series have isomorphic refinements.

Supersolvable Groups 26. Prove that any supersolvable group is countable. 27. Prove that if KÎR is supersolvable and R is cyclic, then K is supersolvable. 28. Prove that a group is supersolvable if and only if it has a series in which each factor group is cyclic of prime order or cyclic of infinite order. Hint: Recall that E « F and F ü K implies E ü K . 29. Let K be supersolvable. Prove that if L is a maximal subgroup of K , then ÐK À LÑ is prime. Hint: First assume that L – K and look at KÎL . Then assume that L is not normal in K and factor by the normal interior L ‰ . Conclude that it is sufficient to consider L ‰ œ Ö"×. Consider the subgroup E in the first step Ö"×  E in a normal cyclic series and how it interacts with L . 30. Prove that if K is supersolvable, then Kw is nilpotent. Hint: Let Ö"× œ K!  â  K8 œ K be a normal cyclic series for K and consider the series Ö"× œ K! ∩ Kw  â  K8 ∩ Kw œ Kw which is normal and cyclic as well. Let F œ K5" and E œ K5 . To show that the series is central, it is sufficient to show that \ EÐF ∩ Kw Ñ EKw ³ Ÿ ^Œ E E E But \ÎE is cyclic and normal in KÎE. What can be said about ÐKÎEÑw ?

Radicals and Residues Definition Let ^ be a class of groups. Let K be a group. 1) If the partially ordered set ^ÐKÑ œ ÖL ü K ± L − ^× has a top element, it is called the ^-radical for K and is denoted by b^ ÐKÑ. 2) If the partially ordered set KÎ^ œ ÖL ü K ± KÎL − ^× has a bottom element, it is called the ^-residue for K and is denoted by b^ ÐKÑ.…

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Fundamentals of Group Theory

31. Show that the ^-radical b^ ÐKÑ and the ^-residue b ^ ÐKÑ are characteristic subgroups of K (if they exist). 32. Let ^ be a class of groups closed under subgroup, quotient and join if at least one factor is normal. Let K be a group and let L ü K . Assume that all mentioned radicals and residues exist. Prove the following: a) If L Ÿ b^ ÐKÑ, then b^ ÐKÑ K Ÿ b^ Œ L L and the inclusion may be proper. b) If L Ÿ b ^ ÐKÑ, then b^ Œ

K L

œ

b ^ ÐKÑ L

33. Let ^ be a class with the extension property: R − ^ß KÎR − O implies K − ^. Prove that the following hold for any group K: a) The ^-radical of KÎb^ ÐKÑ is trivial, that is, b^ Œ

K b^ ÐKÑ

œ Öb^ ÐKÑ×

b) The ^-residue of b ^ ÐKÑ is b ^ ÐKÑ, that is, b^ Ðb^ ÐKÑÑ œ b^ ÐKÑ 34. Let ^ be the class of finite groups. a) Show that there are groups with no ^-radical. b) Show that there are groups with no ^-residue. 35. Let K be a nontrivial finite group. Prove that the following are equivalent: a) K is solvable. b) For every proper normal subgroup O – K, the factor group KÎO has a nontrivial :-radical b: ÐKÎOÑ for some prime :, that is, KÎO has a nontrivial normal :-subgroup. c) For every nontrivial characteristic subgroup O of K, the ; -residue b ; ÐOÑ of O is proper in O for some prime ; , that is, there is a proper normal subgroup E of O such that OÎE is a nontrivial a ; -group.

Additional Problems 36. Let K be a group. Let > œ >K . 5 a) Prove that KÐ5Ñ Ÿ >K ÐKÑ. b) Prove that derlenÐKÑ Ÿ nilclassÐKÑ. 37. Let K be a group. Let > œ >K . Prove the following: a) Ò>5 ÐKÑß >4 ÐKÑÓ Ÿ >54" ÐKÑ Hint: Use induction. Use the three subgroups lemma on Ò>5" ÐKÑß >4" ÐKÑÓ œ ÒÒ>5 ÐKÑß KÓß >4" ÐKÑÓ. b) >5>4 ÐKÑ ÐKÑ Ÿ >54" ÐKÑ for 5ß 4   ".

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317

c) Ò>5 ÐKÑß ' 4 ÐKÑÓ Ÿ ' 45" ÐKÑ for 4   5  ". d) KÐ5Ñ Ÿ >j5 ÐKÑ, where j5 œ #5  ". Hence, derlenÐKÑ is less than or equal to the smallest integer 5 for which j5   - ³ nilclassÐKÑ and so derlenÐKÑ Ÿ ilog# Ð-  "Ñj

Chapter 12

Free Groups and Presentations

Throughout this chapter, \ denotes a nonempty set of formal symbols and \ " denotes the set of formal symbols ÖB" ± B − \×. Further, we assume that \ and \ " are disjoint and write \ w œ \ “ \ " .

Free Groups The idea of a free group J\ on a nonempty set \ is that J\ should be the “most general” possible group containing \ , that is, the elements of \ should generate J\ but have no relationships within J\ . In this case, \ is referred to as a set of free generators or a basis for the free group J\ . To draw an analogy, if Z is a vector space, then a subset U of Z is a basis for Z if and only if for any vector space [ and any assignment of vectors in [ to the vectors in U , there is a unique linear transformation from Z to [ that extends this assignment. This property and the analogous property that defines free groups are best described using univerality. Definition Let \ be a nonempty set. A pair ÐJ ß ,À \ Ä J Ñ where J is a group has the universal mapping property for \ (or is universal for \ ) if, as pictured in Figure 12.1, for any function 0 À \ Ä K from \ to a group K, there is a unique group homomorphism 70 À J Ä K for which 70 ‰ , œ 0 The map 70 is called the mediating morphism for 0 . In this case, we say that 0 can be factored through , or that 0 can be lifted to J . The group J is called a free group on \ and \ is called a set of free generators or a basis for J . The map , is called the universal map for the pair ÐJ ß ,Ñ. We use the notation J\ to denote a free group on \ .…

S. Roman, Fundamentals of Group Theory: An Advanced Approach, DOI 10.1007/978-0-8176-8301-6_12, © Springer Science+Business Media, LLC 2012

319

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Fundamentals of Group Theory

κ

X

F

∃!τf

f

G Figure 12.1 It is clear that the universal map , is injective. Moreover, ,\ generates J\ , for if Ø,\Ù  J\ , then Theorem 4.17 implies that there are distinct group homomorphisms 5ß 7 À J\ Ä K into some group K that agree on Ø,\Ù. Therefore, if 0 œ 5l\ œ 7 l\ , then the uniqueness condition of mediating morphisms is violated. Hence, ,\ generates J . For these reasons, it is common to suppress the map , and think of \ as a subset of J\ . The following theorem says that the universal property characterizes groups up to isomorphism. Theorem 12.1 Let \ be a nonempty set. If ÐJ ß ,Ñ and ÐKß -Ñ are universal for \ , then there is an isomorphism 5 À J ¸ K connecting the universal maps, that is, for which 5‰, œProof. There are unique mediating morphisms 7- À J Ä K and 7, À K Ä J for which 7- ‰ , œ -

and

7, ‰ - œ ,

7- ‰ 7, ‰ - œ -

and

7, ‰ 7- ‰ , œ ,

and so

But the identity maps +K and +J are the unique mediating morphisms for which +K ‰ - œ -

and

+J ‰ , œ ,

and so 7- ‰ 7, œ + K

and 7, ‰ 7- œ +J

which shows that the maps 7- and 7, are inverse isomorphisms.… Definition Let \ be a nonempty set. A word AÐB" ß á ß B8 Ñ over \ w (or the equation AÐB" ß á ß B8 Ñ œ ") is a law of groups if AÐ+" ß á ß +8 Ñ œ " for all groups K and all +3 − K.…

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The following theorem says that what's true in a free group is true in all groups. Theorem 12.2 Let \ be a nonempty set and let AÐB" ß á ß B8 Ñ be a word over \ w . If ÐO\ ß ,Ñ is universal on \ , then the following are equivalent: 1) AÐ,B" ß á ß ,B8 Ñ œ " in J\ 2) AÐB" ß á ß B8 Ñ is a law of groups.… Proof. Let K be a group and let +3 − K for 3 œ "ß á ß 8. The function sending B3 to +3 can be lifted to a unique homomorphism 5À J\ Ä K for which 5,B3 œ +3 and so AÐ+" ß á ß +8 Ñ œ 5AÐ,B" ß á ß ,B8 Ñ œ 5" œ "

…

Cauchy's theorem says that any group is isomorphic to a subgroup of a symmetric group. There is an analog for quotients of free groups, but first we require a definition. Definition Let Y œ ÖK3 ± 3 − M× be a family of groups. A subgroup O of the direct product } Y is called a subdirect product of the family Y if the restricted projection maps 33 lO À O Ä K3 are surjective for all 3 − M .… If K is a group, then the identity map +À K Ä K can be lifted to an epimorphism 5À JK q» K and so K is isomorphic to a quotient of the free group JK . More generally, if \ is a set for which cardÐ\Ñ   cardÐKÑ, then any surjection 0 À \ q» K can be lifted to an epimorphism 5À J\ q» K and the induced map 5À J\ ÎO ¸ K shows that K is isomorphic to a quotient group of the free group J\ . Moreover, we have freedom to choose the values of 7 ÐBOÑ for B − \ arbitrarily, but O depends on that choice. More generally, if Y œ ÖK3 ± 3 − M× is a nonempty family of groups and if \ is a set for which cardÐ\Ñ   cardÐK3 Ñ for all 3 − M , then there are isomorphisms 73 À J\ ÎO3 ¸ K3 for all 3 − M , where 73 ÐBO3 Ñ can be specified arbitrarily, but O3 depends on that choice. Now, the “Chinese” map 5À J\ Ä } J\ ÎO3 3−M

defined by 5ÐAÑÐ3Ñ œ AO3 for A − J\ has kernel M œ +O3 . Hence, if 5 is the induced embedding, then the composition }73 ‰ 5À J\ ÎM ä } 3−M K3 shows that J\ ÎM is isomorphic to a subdirect product of the family Y . Moreover, we can specify that the element BM be sent to any element of } K3 , for all B − \ (again at the expense of O3 ). Theorem 12.3 Let K be a group, let Y œ ÖK3 ± 3 − M× be a nonempty family of groups and let \ be a set.

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1) If cardÐ\Ñ   cardÐKÑ, then there is an isomorphism 7 À J\ ÎR ¸ K where we can choose the elements 7B for B − \ arbitrarily, but R depends on that choice. 2) Suppose that cardÐ\Ñ   cardÐK3 Ñ and that we have specified isomorphisms 73 À K3 ¸ J\ ÎO3 for all 3 − M . Let M œ +O3 . Then J\ ÎM is isomorphic to a subdirect product of the family Y and the isomorphisms 73 can be used to specify the elements BM for B − \ arbitrarily in } K3 , but M depends on that choice.…

Construction of the Free Group The notion of a free group can be defined constructively, without appeal to the universal mapping property. When a constructive approach is taken, one usually hastens to verify the universal mapping property, since this is arguably the most useful property of free groups. On the other hand, since we have chosen to define free groups via universality, we should hasten to give a construction for free groups. Let [ œ Ð\ w ч be the set of all words over the alphabet \ w . As a shorthand, we allow the use of exponents, writing Ú BâB î Ý Ý Ý 8 factors B8 œ Û ðóñóò B" âB" Ý Ý Ý 8 factors Ü%

if 8  ! if 8  ! if 8 œ !

where % is the empty word. It is important to keep in mind that this is only a shorthand notation. Thus, for example, B% B# and B' are both shorthand for BBBBBB and so B% B# œ B' . However, B% B# is shorthand for BBBBB" B" but B# is shorthand for BB and so B% B# Á B# . Since the operation of juxtaposition on [ is associative and since the empty word % is the identity, the set [ is a monoid under juxatpostion. In an effort to form a group, we also want to require that BB" œ % œ B" B for all B − \ . More specifically, consider the following rules that can be applied to members of [ : 1) Removal rules: For =ß . − [ and B − \ , =BB" . Ä =. =B" B. Ä =. BB" Ä % B" B Ä % where one of = or . may be missing.

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2) Insertion rules: For =ß . − [ and B − \ , =. Ä =BB" . =. Ä =B" B. % Ä BB" % Ä B" B where one of = or . may be missing. Let us refer to a finite sequence =" ß á ß =5 of applications of these rules as a reduction of ? to @ of length 5 (even though @ may have greater length than ?). The trivial reduction is an application of no rules and so ? is obtained from itself by the trivial reduction. Since the removal and insertion rules come in inverse pairs, reduction defines an equivalence relation ´ on [ . Let [ Î ´ denote the set of equivalence classes of [ , with ÒAÓ denoting the equivalence class containing A. Since equivalent words must represent the same group element, it is really the equivalence classes that are the candidates for the elements of the free group J\ on \ . Moreover, since ? ´ <ß

@´=

Ê

?@ ´ <=

the equivalence relation ´ is a monoid congruence relation on [ and so we may raise the operation of juxtaposition from [ to [ Î ´ , that is, the operation Ò?ÓÒ@Ó œ Ò?@Ó is well-defined on [ Î ´ and makes [ Î ´ into a group, with identity Ò%Ó and for which /" 5 ÒB/"" âB/55 Ó" œ ÒB/ 5 âB" Ó

However, since it is easier to work with elements of [ rather than equivalence classes, we prefer to use a system of distinct representatives for [ Î ´ . A desirable choice would be the set consisting of the unique word of shortest length from each equivalence class, and so we must prove that such words exist. Let us say that a word A is reduced if it is not congruent to a word of shorter length. It is clear that a removal rule can be applied to a word A − [ if and only if A contains a subword of the form BB" or B" B for B − \ . Further, a reduced word A has no such subword and so no removal rules can be applied to A. We want to prove that the converse also holds, that is, a word A is reduced if and only if no removal rules can be applied to A. Then we can show that the set of reduced words form a system of distinct representatives for [ Î ´ .

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Theorem 12.4 Let < − [ be a word that does not contain a subword of the form BB" or B" B for B − \ . If A ´ <, then there is a reduction from A to < that involves only removal rules. 1) A word is reduced if and only if it does not contain a subword of the form BB" or B" B for B − \ . 2) A word is reduced if and only if it can be written in the form B/"" âB/88 with B3 Á B3" and /3 Á ! for all 3. 3) The set of reduced words is a system of distinct representatives for [ Î ´ . Proof. Among all reductions from A to <, select a reduction with the fewest number of steps and suppose that there is at least one insertion step. Denote the steps by =" ß =# ß á ß =7 and suppose that step =5 results in the word ?5 . Let =5 be the last insertion step, say ?5" œ α" ?5 œ αÐBB" Ñ" where we have marked B with an overbar to distinguish it from any other occurrences of the symbol B. (A similar argument will work for the insertion of B" B.) Since there are no further insertion steps, the pair BB" is never separated during the remaining steps, but must be altered at some point by a subsequent removal rule. Suppose that BB" is unaltered until step =54 and so the intermediate steps are ?5" œ α" ?5 œ αÐBB" Ñ" ?5+1 œ α" ÐBB" Ñ"" ã ?54" œ α4" ÐBB" Ñ"4" There are three possibilities for step =54 . First, if BB" is removed, that is, if ?54 œ α4" "4" then the insertion (and subsequent deletion) of BB" could have been omitted from the reduction process, in which case steps =5 and =54 do nothing can be removed from the reduction, resulting in a shorter reduction, which is a contradiction. The other possibilities involve an interaction of BB" with either α4" or "4" . w One possibility is that α4" œ α4" B" and ?54" œ αw4" B" ÐBB" Ñ"4" ?54 œ αw4" ÐB" Ñ"4" But since ?54 œ α4 "4 , step =54 can be replaced by the removal of BB" ,

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w resulting in the same reduction as in the first case. Similarly, if "4" œ B"4" and w ?54" œ α4" ÐBB" ÑB"4" w ?54 œ α4" ÐBÑ"4"

then since ?54 œ α4" "4" , again we can replace this reduction by the first reduction. Hence, a shortest reduction of A to < cannot have any insertion steps. For part 1), suppose that no removal rules can be applied to < − [ and that A ´ <. Then there is a reduction from A to < that involves only removal steps and so lenÐ<Ñ Ÿ lenÐAÑ. Hence, < is reduced. The converse is clear. For part 2), if A œ B/"" âB/88 is reduced but B3 œ B3" where /3 and /3" have opposite signs, then we can apply a removal rule to A to produce a shorter equivalent word, which is not possible. Thus, if B3 œ B3" , we may add the corresponding exponents. Conversely, if A œ B/"" âB/88 is a word for which B3 Á B3" and /3 Á !, then no removal rule can be applied to A and so part 1) implies that A is reduced. For part 3), any A − [ can be reduced using only removal rules to a word < for which no removal rules apply and so < is reduced. Thus, every word is congruent to a reduced word. Moreover, if ? Á @ are congruent reduced words, then there must be a reduction consisting of zero or more removal steps that brings ? to @, but no removal steps can be applied to ? and so ? œ @.… Thus, each word A − [ is equivalent to a unique reduced word =< and we may use the bijection ÒAÓ Ç A< to transfer the group structure from [ Î ´ to the set J\ of reduced words, specifically, if ?ß @ − J\ , then ?@ œ Ð?@Ñ< It will be convenient to refer to the set V\ of reduced words on \ w by the following name (which is not standard terminology). Definition Let \ be a nonempty set. The concrete free group V\ on \ is the set V\ œ Ö%× ∪ ÖB/"" âB/88 ± B3 Á B3" − \ß /3 Á !× of all reduced words over the alphabet \ w , under the operation of juxtaposition followed by reduction. The rank rkÐV\ Ñ of V\ is the cardinality of \ .… Note that the use of the term “concrete” is not standard. Most authors would refer to V\ simply as the free group on \ , which is justified by the following.

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Theorem 12.5 Let J\ be the concrete free group on a set \ and let 4À \ Ä J\ be the inclusion map. Then the pair ÐJ\ ß 4Ñ is universal for \ and so J\ is a free group on \ . Proof. Let 0 À \ Ä K. If 7 À J\ Ä K is defined by 7% œ " and 7ÐB/"" âB/88 Ñ œ 0 ÐB" Ñ/" â0 ÐB8 Ñ/8 for B/"" âB/88 − J\ , then 7 ‰ 4 œ 0 on \ . Also, it is clear that reduction can take place before or after application of 7 without affecting the final result. However, reduction has no effect in K and so if ? ‡ @ denotes the operation of J\ , then 7 Ð? ‡ @Ñ œ 7 ÐÐ?@Ñ< Ñ œ Ò7 Ð?@ÑÓ< œ 7 Ð?@Ñ œ 7 Ð?Ñ7 Ð@Ñ which shows that 7 is a mediating morphism for 0 . As to uniqueness, if 7 w ‰ 4 œ 7 ‰ 4, then 7 w and 7 agree on \ , which generates J\ and so 7 w œ 7 .…

Relatively Free Groups Freedom can also come in the context of some restrictions. For example, the free abelian group on a set \ is the most “universal” abelian group generated by \ . More generally, if ^ is any class of groups, we can ask if there is a most universal ^-group generated by a nonempty set \ . If so, such a group is referred to as a free ^-group or a relatively free group. For the class ^ of abelian groups, free ^-groups do exist, but this is not the case for all classes of groups, as we will see. The definition of a free ^-group generalizes that of a free group. Definition Let ^ be a class of groups and let \ be a nonempty set. A pair ÐO\ ß ,À \ Ä O\ Ñ where O\ is a ^-group generated by ,\ , has the ^universal property mapping for \ (or is ^-universal for \ ) if for any function 0 À \ Ä K from \ to a ^-group K, there is a unique group homomorphism 70 À O\ Ä K for which 70 ‰ 4 œ 0 The map 70 is called the mediating morphism for 0 . The group O\ is called a free ^-group on \ with free generators or basis \ and , is called the ^universal map for the pair ÐO\ ß ,Ñ.… Note that if ÐO\ ß ,Ñ is ^-universal, then the ^-universal map , is injective. For this reason, one often thinks of \ as a subset of O\ . Proof of the following theorems is left to the reader. Theorem 12.6 Let \ be a nonempty set. If ÐOß ,Ñ and ÐKß -Ñ are ^-universal for \ , then there is an isomorphism 5À O ¸ K for which 5‰, œ-

…

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Definition Let ^ be a class of groups and let \ be a nonempty set. A word AÐB" ß á ß B8 Ñ over \ w (or the equation AÐB" ß á ß B8 Ñ œ ") is a law of ^groups if AÐ+" ß á ß +8 Ñ œ " for all ^-groups O and all +3 − O . If ÖB" ß B# ß á × © \ , then we denote the set of all laws of ^-groups in J\ by _Ð^Ñ or _\ Ð^Ñ.… Theorem 12.7 Let ^ be a class of groups and let \ be an infinite set. Then _\ Ð^Ñ is a fully invariant subgroup of the free group J\ . Proof. It is clear that the product of two laws of ^-groups is a law of ^-groups, as is the inverse of a law of ^-groups. Also, if 5 − EndÐJ\ Ñ, then for any AÐB" ß á ß B8 Ñ − _Ð^Ñ, 5AÐB" ß á ß B8 Ñ œ AÐ5B" ß á ß 5B8 Ñ − _Ð^Ñ and so _Ð^Ñ is fully invariant in J\ .… Theorem 12.8 Let \ be a nonempty set and let AÐB" ß á ß B8 Ñ be a word over \ w . If ÐO\ ß ,Ñ is ^-universal on \ , then the following are equivalent: 1) AÐ,B" ß á ß ,B8 Ñ œ " in O\ 2) AÐB" ß á ß B8 Ñ is a law of ^-groups.… We have said that there are classes of groups for which relatively free groups do not exist. For example, the class of all finite groups is such a class, as we will see later. There is one very important type of class for which relatively free groups do exist, however. Definition Let \ be a nonempty set and let _ œ ÖA3 ± 3 − M× be a subset of the concrete free group J\ . The equational class (or variety) with laws _ is the class X Ð_Ñ of all groups K for which each A3 − _ is identically " on K.… Note that an equational class is closed under subgroup, quotient and direct product. For equational classes, we can construct relatively free groups using our construction of the concrete free group. Theorem 12.9 Let ^ œ X Ð_Ñ be an equational class of groups with laws _. 1) For any group K, the verbal subgroup _ÐKÑ œ ØAÐ+" ß á ß +8 Ñ ± A − _ß +3 − KÙ is fully invariant in K. 2) If J\ is the concrete free group on \ , then the pair ÐO\ œ J\ Î_ÐJ\ Ñß , œ 1 ‰ 4Ñ

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where 1 is projection modulo _ÐJ\ Ñ and 4À \ Ä J\ is inclusion, is ^universal for \ . We refer to ^\ as the concrete ^-free group. Proof. Write J œ J\ . For part 1), if 5 − EndÐKÑ, then for any +3 − K , 5AÐ+" ß á ß +8 Ñ œ AÐ5+" ß á ß 5+8Ñ − _ÐKÑ and so _ÐKÑ is fully invariant in K . For part 2), to see that O œ O\ is a ^group, if AÐB" ß á ß B8 Ñ − _, then for any +3 − J , AÐ+" _ÐJ Ñß á ß +8 _ÐJ ÑÑ œ AÐ+" ß á ß +8Ñ_ÐJ Ñ œ _ÐJ Ñ and so O\ satisfies the laws in _. To see that Ð^\ ß 1 ‰ 4Ñ is ^-universal, referring to Figure 12.2, let 0 À \ Ä L , where L is a ^-group.

j

X

FX

π

σ

f

KX=F/L(F) τ

H Figure 12.2 Then 0 can be lifted uniquely to a homomorphism 5À J Ä L satisfying 5 ‰ 4 œ 0 . But if AÐB" ß á ß B8 Ñ − _ and ?3 − J\ , then 5AÐ?" ß á ß ?8 Ñ œ AÐ5?" ß á ß 5?8 Ñ œ " Hence, AÐ?" ß á ß ?8 Ñ − kerÐ5Ñ and so _ÐJ Ñ Ÿ kerÐ5Ñ. Thus, the universality of quotients implies that 5 can be lifted uniquely to a homomorphism 7 À J Î_ÐJ Ñ Ä L for which 7 ‰ 1 œ 5 and so 7 ‰1‰4œ5‰4œ0 As to uniqueness, if 7 w ‰ 1 ‰ 4 œ 7 ww ‰ 1 ‰ 4 œ 0 then the uniqueness of 5 implies that 7 w ‰ 1 œ 7 ww ‰ 1 and the uniqueness of 7 implies that 7 w œ 7 ww .…

More on Equational Classes If ^ is a class of groups, then the equational class X Ð_Ð^ÑÑ is the class of all groups that satisfy the laws of ^-groups. Of course, a ^-group satisfies the laws of ^-groups and so ^ © X Ð_Ð^ÑÑ. The interesting issue is that of equality, that is, for which classes ^ is it true that the laws of ^-groups hold only for ^groups? The answer is that this happens if and only if ^ is an equational class.

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For if ^ œ X Ð`Ñ is the equational class for a set ` of laws over \ w , then ` © _Ð^Ñ and so X Ð_Ð^ÑÑ © X Ð`Ñ œ ^ © X Ð_Ð^ÑÑ whence X Ð_Ð^ÑÑ œ ^. Theorem 12.10 A class ^ of groups satisfies ^ œ X Ð_Ð^ÑÑ that is, the laws of ^-groups hold only for ^-groups, if and only if ^ is an equational class.… Example 12.11 Let ^ be the class of all finitely-generated groups. If AÐB" ß á ß B8 Ñ is a law of ^, then AÐB" ß á ß B8 Ñ œ " in all finitely-generated groups and therefore in all groups. Hence, X Ð_Ð^ÑÑ is the class of all groups and so ^ is not an equational class.… The following theorem makes it relatively easy to tell when a class ^ is an equational class. Theorem 12.12 (Birkhoff) The following are equivalent for a class ^ of groups: 1) ^ is an equational class 2) ^ is closed under subgroup, quotient and direct product 3) ^ is closed under quotient and subdirect product. Proof. It is clear that 1) implies 2), which implies 3). To show that 3) implies 1), we show that X Ð_Ð^ÑÑ © ^. Let K − X Ð_Ð^ÑÑ, that is, K satisfies the laws of ^-groups. For the proof, we work with quotients of free groups. Now, if cardÐ] Ñ   cardÐKÑ, there is an R ü K for which K ¸ J] ÎR and so J] ÎR satisfies the laws of ^-groups. Hence, if AÐC" ß á ß C8 Ñ − _] Ð^Ñ, then AÐC" ß á ß C8 Ñ − R and so _] Ð^Ñ Ÿ R . Hence, K¸

J] J] R ¸ ‚ R _] Ð^Ñ _] Ð^Ñ

and the proof would be complete if we knew that J] Î_] Ð^Ñ was a ^-group. In fact, the same argument works for any M Ÿ _] Ð^Ñ, since then M Ÿ R and so K¸

J] J] R ¸ ‚ R M M

Thus, we only need to find an M Ÿ _] Ð^Ñ for which J] ÎM is a ^-group. Now, if A − J] is not a law of ^-groups, then there is a ^-group OA that violates A, that is, for which

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Fundamentals of Group Theory

AÐ5Aß" ß á ß 5Aß8A Ñ Á " for some 5Aß3 − OA . Moreover, if cardÐ] Ñ   cardÐOA Ñ, then there is an isomorphism 7A À J] ÎRA ¸ OA for which 7A ÐC3 RA Ñ œ 5Aß3 for all 3. Hence, J] ÎRA also violates A in the sense that AÐC" ß á ß C8A Ñ Â RA and so

M ³ , RA Ÿ _] Ð^Ñ AÂ_Ð^Ñ

But Theorem 12.3 implies that the quotient J] ÎM is isomorphic to a subdirect product of Y and is therefore a ^-group.… We can now relate equational classes and existence of relatively free groups. Theorem 12.13 (Birkhoff) The following are equivalent for a nontrivial class ^ of groups: 1) ^ is an equational class, that is, X Ð_Ð^ÑÑ œ ^ 2) ^ is closed under quotient and for every nonempty set \ , there is a ^universal pair ÐO\ ß ,Ñ. Proof. We have seen that 1) implies 2). For the converse, we show that 2) implies X Ð_Ð^ÑÑ © ^. Let K − X Ð_Ð^ÑÑ.

j

G

κ

FG

σ

λ

G

τ

KG Figure 12.3 Referring to Figure 12.3, let 4À K Ä JK be the inclusion map into the concrete free group JK . We lift two maps. The ^-universal map , can be lifted to a unique homomorphism -À JK Ä OK for which -4 œ ,. Moreover, - is surjective since ,K generates OK . Also, the identity map +À K Ä K can be lifted to a unique epimorphism 5À JK q» K for which 54 œ +. To see that kerÐ-Ñ Ÿ kerÐ5Ñ, if AÐB" ß á ß B8 Ñ is a word over \ w and if AÐ4+" ß á ß 4+8 Ñ − kerÐ-Ñ for +3 − K, then AÐ,+" ß á ß ,+8 Ñ œ -AÐ4+" ß á ß 4+8 Ñ œ " in OK and so Theorem 12.8 implies that AÐB" ß á ß B8 Ñ − _Ð^Ñ. Hence in K ,

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5AÐ4+" ß á ß 4+8 Ñ œ AÐ54+" ß á ß 54+8 Ñ œ AÐ+" ß á ß +8Ñ œ " and so AÐ4+" ß á ß 4+8 Ñ − kerÐ5Ñ, whence kerÐ-Ñ Ÿ kerÐ5Ñ. Hence, 5 induces an epimorphism 7À OK q» K defined for each 5 − OK by 7 5 œ 5-" Ð5Ñ and so K ¸ OK ÎkerÐ7 Ñ and since the latter is a ^-group, so is K.…

Free Abelian Groups The class of abelian groups is an equational class, with law ÒBß CÓ œ ". Thus, free abelian groups exist. In fact, if K is a group, then the verbal subgroup is _ÐKÑ œ ØÒ+ß ,Ó ± +ß , − KÙ which is just the commutator subgroup Kw of K. Hence, Theorem 12.9 implies that if J\ is the concrete free group on \ , then the group E\ œ J\ ÎJ\w is free abelian. Definition Let \ be a nonempty set. If J\ is the concrete free abelian group on \ , then E\ œ J\ ÎJ\w is the concrete free abelian group on \ . It is customary to think of E\ as the group J\ with the additonal condition that the elements of \ commute.… Theorem 12.14 If \ is a nonempty set, then the free abelian group E\ satisfies E\ ¸ { ØBÙ ¸ { ™ B−\

B−\

Proof. The function 0 À \ Ä { ØBÙ defined by 0 ÐBÑÐCÑ œ œ

B "

if C œ B if C Á B

for all C − \ can be lifted uniquely to a homomorphism 7 À E\ Ä { ØBÙ for which 7ÐB/"" âB/88 Ñ œ 0 ÐB" Ñ/" â0 ÐB8 Ñ/8 and so 7ÐB/"" âB/88 ÑÐCÑ œ œ

B/55 "

if C œ B5 if C  ÖB" ß á ß B8 ×

Now, 7 is surjective, since if α − { ØBÙ has support ÖB" ß á ß B8 × and αÐB5 Ñ œ B/55 , then α œ 7 ÐB/"" âB/88 Ñ. Also, 7 is injective, since if 7 ÐB/"" âB/88 Ñ œ !, then B/55 œ " for all 5 and so /5 œ ! for all 5 . Thus, 7 is an isomorphism.…

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The following theorem explains the term basis used for the set \ in the free abelian group E\ . Definition A subset W of the free abelian group E\ is independent in E\ if =3 − Wß

=/"" â=/88 œ %ß

=3 Á =4

Ê

/3 œ ! for all 3

…

Theorem 12.15 Let \ be a nonempty subset of an abelian group A. The following are equivalent: 1) E is a free abelian group with basis \ 2) \ is independent in E and generates E 3) Except for the order of the factors, every nonidentity element + − E has a unique expression of the form + œ B/"" âB/88 for B3 Á B4 ß /5 Á !ß 8   " Proof. If 1) holds, we have seen that \ generates E and if A œ B/"" âB/88 œ % for B3 Á B4 , then /3 œ ! for all 3, since otherwise A would be a reduced word equivalent to the shorter word %. Hence, 2) holds. If 2) holds, then every element of E has at least one such expression. But if + − E has two distinct expressions: 07 + œ B/"" âB/88 œ C"0" âC7

where we may assume that B8 Á C7 (or we can cancel), then 07 B/"" âB/88 C7 âC"0" œ %

violates independence. Hence, 3) holds. Finally, to see that 3) implies 1), suppose that 0 À \ Ä K, where K is an abelian group. If a mediating morphism 7 does exist, then 7ÐBÑ œ 0 ÐBÑ for all B − \ and so 7 is unique, since \ generates E. Define a map 7 À E Ä K by 7ÐB/"" âB/88 Ñ œ 0 ÐB" Ñ/" â0 ÐB8 Ñ/8 which is well defined since the expressions B/"" âB/88 are unique up to order but K is abelian. The map 7 is easily seen to be a homomorphism.… The next theorem says that, up to isomorphism, there is only one free (or free abelian) group of each cardinal rank. Theorem 12.16 Let \ and ] be nonempty sets. 1) a) E\ ¸ E] Í k\ k œ k] k b) If W generates E\ , then kW k   k\ k 2 ) a ) J \ ¸ J ] Í k\ k œ k] k

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b) If W generates J\ , then kW k   k\ k Proof. Suppose first that k\ k œ k] k and let 0 À \ Ä ] be a bijection. Extend the range of 0 so that 0 À \ Ä J] (or 0 À \ Ä E] ). Then there is a unique mediating morphism 7 À J\ Ä J] (or 7 À E\ Ä E] ) for which 7ÐBÑ œ 0 ÐBÑ for all B − \ . To see that 7 is injective, if A œ B/"" âB/88 and 0 ÐB3 Ñ œ C3 , then 7 ÐAÑ œ 7 ÐB/"" âB/88 Ñ œ 0 ÐB" Ñ/" â0 ÐB8Ñ/8 œ C"/" âC8/8 Hence, 7 ÐAÑ œ % implies /3 œ ! for all 3 and so A œ %. Also, 7 is surjective since 0 À \ Ä ] is surjective. Hence, J\ ¸ J] (and E\ ¸ E] ). For the converse of part 1), let E represent either E\ or E] . We use additive notation. The quotient EÎ#E is elementary abelian of exponent # and is therefore a vector space over ™# . Moreover, if W generates the group E, then WÎ#E œ Ö=  #E ± = − W× generates the vector space EÎ#E over ™# and if W is independent in E, then WÎ#E is linearly independent in EÎ#E, since an equation of the form Ð="  #EÑ  â  Ð=8  #EÑ œ #E for 8  ! and =3 Á =4 in W implies that ="  â  =8 œ #Ð/" >"  â  /7 >7 Ñ for >3 − W and /3 − ™, which is not possible. It follows that \Î#E\ is a basis for E\ Î#E\ and so dimÐE\ Î#E\ Ñ œ k\ k

and similarly for ] Þ Thus, since E\ ¸ E] implies E\ Î#E\ ¸ E] Î#E] , we have k\ k œ dimÐE\ Î#E\ Ñ œ dimÐE] Î#E] Ñ œ k] k

and if W generates E\ , then

kW k   kWÎ#E\ k   k\ k

This completes the proof of part 1). For part 2), if J\ ¸ J] , then E\ ¸ J\ ÎJ\w ¸ J] ÎJ]w ¸ E]

and so part 1) implies that k\ k œ k] k. Finally, if W generates J\ , then

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WÎJ\w œ Ö=J\w ± = − W× generates E\ œ J\ ÎJ\w and so kW k   kWÎJ\w k   k\ÎJ\w k œ k\ k

…

Theorem 12.17 Let E\ be free abelian on \ . Then all independent sets have cardinality at most k\ k. Proof. It is sufficient to prove the result for E œ { B−\ ™B where ™B œ ™ for all B. The set Z œ { B−\ B is a vector space over the rational field  and it is easy to see that a subset U œ Ö@3 ± 3 − M× © E is dependent in E if and only if U is linearly dependent over . But in the vector space Z , all sets of cardinality greater than k\ k are linearly dependent over  and therefore also over ™.… The Nielsen–Schreier Theorem says that every subgroup of a free group is free and so the subgroups of free groups are very restricted. (Nielsen proved this result for finitely-generated groups in 1921 and Schreier generalized it to all groups in 1927.) Theorem 12.18 1) Any subgroup of a free group is free. 2) Any subgroup W of a free abelian group E\ is free abelian and rkÐWÑ Ÿ rkÐEÑ. Proof. We omit the difficult proof of part 1) and refer the interested reader to Robinson [26]. For part 2), we may assume that E\ œ { B−\ ØBÙ and that \ is well ordered. Since the elements of W have finite support, for 0 − W , we can let 3Ð0 Ñ be the largest index B for which 0 ÐBÑ Á ". For each B − \ , consider the set MB œ Ö0 ÐBÑ ± 0 − Wß 3Ð0 Ñ Ÿ B× Then MB Ÿ ØBÙ and so MB œ Ø0B ÐBÑÙ for some 0B − WB . We show that W is free on the set U œ Ö0B ± B − \ß 0B ÐBÑ Á "× If U does not span W , among those elements of W not in the span of U , choose an element 1 for which C œ 3Ð1Ñ is the smallest possible. Since 1ÐCÑ − MC œ Ø0C ÐCÑÙ, it follows that 1ÐCÑ œ 0C5 ÐCÑ for some nonzero 5 − ™. Then Ð10C5 ÑÐBÑ œ 1ÐBÑ0C5 ÐBÑ œ " for all B   C and so 3Ð10C5 Ñ  C, which implies that 10C5 is in the span of U . But then

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1 œ Ð10C5 Ñ0C5 is also in the span of U , a contradiction. Thus, U spans W . Also, U is independent, since if 0B/"" â0B/88 œ " where B3  B4 for 3  4, then applying this to B8 gives 0B/88 ÐB8 Ñ œ " and so /8 œ !. Similarly, /3 œ ! for all 3 and so U is independent. Hence, Theorem 12.15 implies that W is free over U . Also, it is clear that kU k Ÿ k\ k and so rkÐWÑ Ÿ rkÐEÑ.…

Applications of Free Groups Sometimes free groups can help produce complements. Theorem 12.19 If 5À K q» J\ is an epimorphism, where J\ is free on \ , then O œ kerÐ5Ñ is complemented in K. Proof. Define a function 7 À \ Ä K by letting 7 B be a fixed member of 5" ÐBÑ. Since J\ is free on \ , there is a unique homomorphism 7 À J\ Ä K that extends 7 on \ . Since for any B − \ , 5 ‰ 7 ÐBÑ œ B it follows that 5 ‰ 7 œ +. In other words, 7 is a right inverse of 5 and so Theorem 5.23 implies that kerÐ5Ñ is complemented in K .… With the help of free groups, we can provide an example of a finitely-generated nonabelian group with a subgroup that is not finitely generated. We have already proved (Theorem 2.21) that if K is an 8-generated abelian group, then every subgroup of K can be generated by 8 or fewer elements. Theorem 12.20 Let \ œ ÖBß C× and let J\ be the #-generated free group on \ . Let K œ ØWÙ, where W œ ÖC5 BC5 ± 5  !× Then K is isomorphic to the free group J^ on a countably infinite set ^ œ ÖD" ß D# ß á × and so is not finitely generated. Proof. Consider the function 0 À ^ Ä K defined by 0 ÐD5 Ñ œ C5 BC5 . Then there is a unique mediating morphism 7 À J^ Ä K for which 7 ÐD5 Ñ œ C5 BC5 . It is clear that 7 is surjective.

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In addition, if /

/

7 ÐD3"3" âD3737 Ñ œ % where 35 Á 35" , /35 Á ! and 7   ", then C3" B/3" C3" 3# B/3# C3# 3$ B/3$ âB/37" C37" 37 B/37 C37 œ % in K Ÿ J\ . Since the left-hand side can be reduced to % using only removal steps, it follows that 35 œ 35" for all 5 and so 7 œ " and C3" B/3" C3" œ % But the left-hand side of this equation reduces to % by removal steps if and only if /" œ !, which is false. Hence, 7 is injective and therefore an isomorphism.… We can also provide an example of a group K with a subgroup L Ÿ K for which +L+"  L . Theorem 12.21 Let J\ be the free group on \ œ ÖBß C×. Then J\ has a subgroup L for which BLB"  L . Proof. Let L consist of the empty word % and the set of all words of the form A œ B8" C5" B8# C5# âB8< C5< B8<" with <   ", 53 Á !ß 8"   !ß 8<" Ÿ !, 83 Á ! for # Ÿ 3 Ÿ < and 83 œ !. Note that C − L . We leave it to the reader to show that L is a subgroup of J\ . It is clear that BAB" œ B8" " C 5" B8# C 5# âB8< C 5< B8<" " − L and so BLB" Ÿ L . However, BAB" Á C for all A − L since if A Á %, then BAB" has length at least $. Hence, BLB"  L .…

Presentations of a Group One way to define a group is to list all of the elements of the group and then give the group's multiplication table, which shows explicitly how to multiply all pairs of elements of the group. Then it is necessary to verify the defining axioms of a group: associativity, identity and inverses. For example, the cyclic group G$ Ð+Ñ is G$ œ Ö"ß +ß +# × with multiplication table

Free Groups and Presentations

" + +#

" " + +#

+ + +# "

337

+# +# " +

Of course, we generally abbreviate this description by writing G$ œ Ö+3 ± ! Ÿ 3 Ÿ #×ß

+3 +4 œ +Ð34) mod $

It is routine in this case to check the group axioms. On the other hand, it is tempting to define a group by giving a set of generators for the group along with some properties satisfied by these generators. The issue then becomes one of deciding whether there is a group that has these generators with these properties. To illustrate, consider the following description of a group K: K œ Ø+ß ,Ùß +# œ "ß , % œ "ß +,+, œ " This description gives a nonempty set \ œ Ö+ß ,× of generators for K and certain relations on K, that is, equations of the form A œ " where A is a word over the generators and their inverses. Note that there is nothing in the description above that precludes the possibility that + œ , . In order to guarantee that such a group exists, we require that all relations must have the form A œ ". The left-hand side A is referred to as a relator. Expressions such as A Á " are not permitted. However, it is customary to take the liberty of writing a relation in the form A œ @ as a more intuitive version of A@" œ ". For example, the last relation above can be written ,+ œ +" , "

or

,+ œ +, $

In view of the precise nature of relations, given any nonempty generating set \ and any set e of relations, there is always one group that is generated by \ and satisfies the relations e: It is the trivial group, where each generator is taken to be the identity. On the other hand, the best hope for getting a nontrivial group generated by \ and satisfying the relations e is to start with the concrete free group J\ and factor out by the smallest normal subgroup R required in order to satisfy the given relations, that is, the normal closure of the relators of e.

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With respect to the example above, the quotient group K" œ

JÖBßC× ØB# ß C% ß BCBCÙnor

has generators + œ BR and , œ CR , where R œ ØB# ß C% ß BCBCÙnor that satisfy the relations given for a group K. Thus, we are lead to the concept of a free presentation of a group, given in the next definition. Definition Let J\ be free on \ . An epimorphism 5À J\ q» K is called a free presentation of K. The set \ is called a set of generators for the presentation and if kerÐ5Ñ œ ØeÙnor so that K¸

J\ ØeÙnor

the set e is called a set of relators of the presentation. In this case, we write K ¸ Ø\ ± eÙ If < − e is a relator, then the equation < œ " is called a relation.… We will often refer to a free presentation of K simply as a presentation of K. It is common to say that the group K itself has presentation Ø\ ± eÙ when K¸

J\ ØeÙnor

and that K is defined by generators and relations. Note, however, that it is the set 5\ that actually generates K . A presentation Ø\ ± eÙ is finite if \ ∪ e is a finite set. Finally, we will often blur the distinction between a relator and the corresponding relation, using whichever is more convenient at the time. We can form a more concrete version of the group defined by generators \ and relations e as follows. If K œ Ø\Ù and if J\ is the concrete free group on \ , then a word over \ w has two contexts: as an element of J\ and as an element of K. Moreover, there is a unique epimorphism 5À J\ q» K defined by specifying that 5B œ B for all B − \ . This map can be thought of simply as a change of context and it is convenient to give it this name officially. Definition Let K œ Ø\Ù and let J\ be the concrete free group on \ . 1) We call the unique epimorphism 5À J\ q» K defined by 5B œ B for all B − \ the change of context map associated to K and \ . 2) If the change of context map 5À J\ q» K has kernel R œ ØeÙnor , so that the induced map 7 À J\ ÎR Ä K defined by

Free Groups and Presentations

339

7 ÐBR Ñ œ B is an isomorphism, we say that K has concrete presentation Ø\ ± e Ù and write K œ Ø\ ± eÙ.… It is clear that if K œ Ø\ ± eÙ with R œ ØeÙnor and if AÐB" ß á ß B8 Ñ is a word over \ w , then AÐB" ß á ß B8 Ñ œ " in K

Í Í

AÐB" R ß á ß B8 R Ñ œ R in J\ ÎR AÐB" ß á ß B8 Ñ − R in J\

Every Group Has a Presentation The change of context map 5À JK q» K shows that every group has a concrete presentation. Moreover, the kernel of this presentation is essentially the multiplication table for K . To be more specific, if +ß ,ß - − K and - œ +, , then +,- " œ " in K and so +,- " must be factored out of JK . So let R œ Ø+,- " ± +ß ,ß - − Kß - œ +, in KÙ Ÿ JK It is easy to see that R is a fully invariant subgroup of JK and that R Ÿ kerÐ5Ñ. For the reverse inclusion, if AÐ+" ß á ß +8 Ñ − kerÐ5Ñ, then AÐ+" ß á ß +8 Ñ œ " in K and so AÐ+" ß á ß +8 Ñ − R . Thus, R œ kerÐ5Ñ and so K œ ØK ± eÙ, where e œ Ö+,- " ± +ß ,ß - − Kß - œ +, in K× Note also that if K is finite, then so is e . Theorem 12.22 Every group K has concrete presentation ØK ± eÙ, where e œ Ö+,- " ± +ß ,ß - − Kß - œ +, in K× Moreover, if K is finite, then ØK ± eÙ is a finite presentation of K .… The concrete presentation ØK ± eÙ is rather large and we can improve upon it in general. Theorem 12.23 If K ¸ Ø\ ± eÙ, then K œ Ø] ± f Ù, where k] k Ÿ k\ k and kek Ÿ kf k. In particular, K has a finite presentation if and only if it has a finite concrete presentation. Proof. Let .À J\ q» K be a free presentation of K with kernel R œ ØeÙnor . Then the set ] œ .\ generates K. Let 5À J] q» K be the change of context map. If f is the set of relators in ] obtained from e by replacing each

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Fundamentals of Group Theory

occurrence of B − \ by .B, then the following are equivalent: AÐ.B" ß á ß .B8 Ñ − kerÐ5Ñ AÐ.B" ß á ß .B8 Ñ œ " in K AÐB" ß á ß B8 Ñ − ØeÙnor AÐ.B" ß á ß .B8 Ñ − Øf Ùnor and so kerÐ5Ñ œ Øf Ùnor and K œ Ø] ± f Ù.…

Finitely Presented Groups A group is finitely presented if it has a finite presentation. Theorem 12.24 If K has a finite presentation and if \ is a generating set for K , then K has a finite presentation of the form ØB" ß á ß B8 ± <" ß á ß <7 Ù where B3 − \ . Proof. Let Ø] ± f Ù be a finite concrete presentation of K , with ] œ ÖC" ß á ß C? ×

and f œ Ö=3 ÐC" ß á ß C? Ñ ± 3 œ "ß á ß @×

Then there is a finite subset \! œ ÖB" ß á ß B8 × © \ for which ] © Ø\! Ù and so \! generates K and we can write B3 œ 03 ÐC" ß á ß C? Ñà

3 œ "ß á ß 8

C4 œ -4 ÐB" ß á ß B8 Ñà

4 œ "ß á ß ?

and

Let L œ ØB" ß á ß B8 ± eÙ where e is the set of relators formed from the relations =3 Ð-" ÐB" ß á ß B8 Ñß á ß -? ÐB" ß á ß B8 ÑÑ œ "à

3 œ "ß á ß @

B4 œ 04 Ð-" ÐB" ß á ß B8 Ñß á ß -? ÐB" ß á ß B8 ÑÑà

4 œ "ß á ß 8

and

Since each of these relations holds in K, the subgroup ØeÙnor is contained in the kernel of the change of context epimorphism 5À J\! q» K and so 5 induces an epimorphism 5w À L q» K defined by 5w ÐB3 Ñ œ B3 . To see that 5w is injective, if 5w ÐAÐB" ß á ß B8 ÑÑ œ " then

Free Groups and Presentations

341

AÐB" ß á ß B8 Ñ œ " in K, whence AÐ0" ÐC" ß á ß C? Ñß á ß 08 ÐC" ß á ß C? ÑÑ œ " in K and so AÐ0" ÐC" ß á ß C? Ñß á ß 08 ÐC" ß á ß C? ÑÑ − Øf Ùnor in J] . Replacing each C4 by -4 ÐB" ß á ß B8 Ñ implies that AÐB" ß á ß B8 Ñ − ØeÙnor in L , that is, AÐB" ß á ß B8 Ñ œ " in L . Hence, 5w is an isomorphism and so K has presentation ØB" ß á ß B8 ± eÙ.… Theorem 12.25 Let K be a group and let R ü K. If R and KÎR are finitely presented, then so is K. Proof. Let R œ ØB" ß á ß B8 ± <" ß á ß
Í

=3 ÐC" ß á ß C8 Ñ œ A3 ÐB" ß á ß B8Ñ

where A3 ÐB" ß á ß B8 Ñ is a word in the B3 's and B" 3 's. Also, R is normal if and only if C4 B3 C4" œ @3ß4 ÐB" ß á ß B8 Ñ

and C4" B3 C4 œ D3ß4 ÐB" ß á ß B8 Ñ

for words @3ß4 and D3ß4 . The following set e of relations captures the relations <3 and =4 as well as the fact that R ü K : <3 ÐB" ß á ß B8 Ñ œ " for 3 œ "ß á ß ? =3 ÐC" ß á ß C8 Ñ œ A3 ÐB" ß á ß B8 Ñ for 3 œ "ß á ß @ C4 B3 C4" œ @3ß4 ÐB" ß á ß B8 Ñ for 3 œ "ß á ß 8à 4 œ "ß á ß 7 C4" B3 C4 œ D3ß4 ÐB" ß á ß B8 Ñ for 3 œ "ß á ß 8à 4 œ "ß á ß 7 w Let \ w œ ÖBw" ß á ß Bw8 ×, ] w œ ÖC"w ß á ß C7 × and Ew œ \ w ∪ ] w and let w L œ ØEw ± eEÄE wÙ

If R w œ ØBw" ß á ß Bw8 Ù Ÿ L , then the relators in ew show that R w ü L .

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Fundamentals of Group Theory

Let 0 À JEw Ä K be the unique epimorphism for which 0 Bw3 œ B3 and 0 C4w œ C4 . Since Øew Ùnor Ÿ kerÐ0 Ñ, there is a unique epimorphism 1À L q» K for which 1Bw3 œ 0 Bw3 œ B3

and 1C3w œ 0 C3w œ C3

Moreover, the restriction 1À R w q» R is an isomorphism, since if 1ÐAÐBw" ß á ß Bw8 ÑÑ œ " then AÐB" ß á ß B8 Ñ œ " in R and so AÐB" ß á ß B8 Ñ − Øew Ùnor in J\ . It follows that w AÐBw" ß á ß Bw8 Ñ − ØeEÄE w Ùnor

in JEw and so AÐBw" ß á ß Bw8 Ñ œ " in R w . Hence, if O œ kerÐ1Ñ, then O ∩ R w œ Ö"× Our goal is to show that O œ Ö"×. Since 1ÐR w Ñ œ R , the epimorphism 1À L q» K induces an epimorphism 1w À LÎR w q» KÎR for which 1w ÐC4w R w Ñ œ 1ÐC4w ÑR œ C4 R for all 4. Moreover, 1w is an isomorphism, since if w 1w ÐAÐC"w R w ß á ß C7 R w ÑÑÑ œ AÐC" R ß á ß C7 R ÑÑ œ R

in KÎR , then AÐC" R ß á ß C7 R ÑÑ − Ø=" ÐC" R ß á ß C7 R Ñß á ß =? ÐC" R ß á ß C7R ÑÙnor in the free group on ÖC" R ß á ß C7 R ×. Hence, w w w AÐC"w R w ß á ß C7 R w ÑÑ − Ø=" ÐC"w R w ß á ß C7 R w Ñß á ß =? ÐC"w R w ß á ß C7 R w ÑÙnor w in the free group on ÖC"w R w ß á ß C7 R w ×, which implies that w AÐC"w ß á ß C7 Ñ − Rw

that is, w AÐC"w R w ß á ß C7 R w ÑÑ œ R w

in LÎR w . Thus, 1w is an isomorphism. Finally, if " Á 5 − O , then 5  R w and so 5R w Á R w , whence R œ 1Ð5ÑR œ 1w Ð5R w Ñ Á R

Free Groups and Presentations

343

It follows that O œ Ö"×, that is, 1À L ¸ K is an isomorphism and so K is finitely presented.…

Combinatorial Group Theory Before looking at other examples, let us return briefly to the question of whether a given presentation Ø\ ± eÙ defines a nontrivial group. From one point of view, this question has a rather surprising answer. It can be shown that no algorithm can ever exist that determines whether or not an arbitrary set of generators and relations defines a nontrivial group! Nor is there any algorithm that determines whether the group defined by an arbitrary finite presentation is finite or infinite. Definition A decision problem is a problem that has a yes or no answer, such as whether or not a given word in J\ is the identity in a group K . 1) A decision problem is decidable or solvable if there is an algorithm, called a decision procedure, that stops after a finite number of steps and returns “yes” when the answer is yes and “no” when the answer is no. A decision problem is undecidable or unsolvable if it is not decidable. 2) A decision problem is semidecidable or semisolvable if there is an algorithm that stops after a finite number of steps and returns “yes” when the answer is yes. However, the algorithm need not stop if the answer is no.… The word problem for a group K with presentation Ø\ ± eÙ, first formulated in 1911 by Max Dehn, is the problem of deciding whether or not an arbitrary word over \ w is the identity element of K (or, equivalently, whether or not two arbitrary words over \ w are the same element of K). It has been shown that there exist individual groups K with finite presentations for which the word problem is unsolvable. On the other hand, there are large classes of groups for which the word problem is solvable. For example, the word problem is solvable for all free groups (in view of Theorem 12.4), for all finite groups and for all finitely-generated abelian groups. In fact, it is an active area of current research in group theory to study classes of groups for which the word problem can be solved. On the other hand, the word problem for finitely-presented groups is semidecidable. For if Ø\ ± eÙ is a finite presentation of a group K and if A − K, then since e is a finite set, there is an algorithm that checks all of the elements of ØeÙnor one-by-one looking for A. If A œ " in K, then this algorithm will eventually encounter A. The problem is that the algorithm will not terminate if A Á " and so this is not a decision procedure. One way to mitigate this problem is to intermix the steps of this algorithm with the steps of another algorithm that stops if A Á ", assuming that such an algorithm exists.

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Fundamentals of Group Theory

For example, the following defines a class of group for which such an algorithm does exist. Definition A group K is residually finite if for any " Á + − K , there is a normal subgroup R ü K for which +  R and KÎR is finite.… Theorem 12.26 The word problem is solvable for the class of all finitelypresented residually finite groups. Proof. Let K œ Ø\ ± eÙ be a finite presentation of a residually finite group K and let A − K. It is possible to enumerate all finite groups by constructing multiplication tables. Also, for a given finite group J , there are only a finite number of group homomorphisms from K to J , since all such homomorphisms are uniquely determined by the functions 0 À \ Ä J . Consider the following algorithm: 1) Compute the next finite group J . 2) For each group homomorphism 5À K Ä J , stop the algorithm if 5A Á ". Now, if A Á " in K, then since K is residually finite, there is an R ü K for which A  R and KÎR is finite. Hence, the canonical projection 1À K Ä KÎR , which is encountered in step 2) above, satisfies 1A Á " and so the algorithm will stop if A Á ". Thus, we can intermix this algorithm with the aforementioned algorithm to get a decision procedure for the word problem for K.… The issues discussed above fall under the auspicies of an area of algebra known as combinatorial group theory.

On the Order of a Presented Group Since a relation cannot be a nonequality, there is no way to specify the order of an element or subgroup of a group by relations. Thus, for example, among the following descriptions of a group, only the first description is a presentation: 1) K" œ Ø+ß , ± +# œ "ß , % œ "ß +,+, œ "Ù 2) K# œ Ø+ß ,Ù, 9ÐK# Ñ œ )ß +# œ "ß , % œ "ß +,+, œ " 3) K$ œ Ø+ß ,Ùß 9Ð+Ñ œ #ß 9Ð,Ñ œ %ß +,+, œ " Since the relation +,+, œ " is equivalent to the commutativity relation ,+ œ +, $ each of the groups above has underlying set W œ Ö+3 , 4 ± ! Ÿ 3 Ÿ "ß ! Ÿ 4 Ÿ $×

Free Groups and Presentations

345

and so 9ÐK5 Ñ Ÿ ) for 5 œ "ß #ß $. Moreover, since 9ÐK# Ñ œ ) implies 9Ð+Ñ œ # and 9Ð,Ñ œ %, any group satisfying 2) also satisfies 3). Conversely, 9ÐK$ Ñ œ % or ), but if 9ÐK$ Ñ œ %, then K$ œ Ø,Ù and + œ , # , which does not satisfy +,+, œ ". Hence, 9ÐK$ Ñ œ ) and so 2) and 3) describe the same group. Thus, if we show that a group K fitting description 2) or 3) exists, then K has order ) and satisfies the relations e given by 1). Hence, e is contained in the kernel of the change of context map 5À JÖ+ß,× Ä K and so )   ÐJÖ+ß,× À ØeÙnor Ñ   ÐJÖ+ß,× À kerÐ5ÑÑ œ kK k œ )

It follows that kerÐ5Ñ œ ØeÙnor and so K has presentation Ø\ ± eÙ. Theorem 12.27 Suppose that a group with presentation Ø\ ± eÙ has order at most 8  ∞. Then any group K of order 8 generated by \ and satisfying the relations in e has presentation Ø\ ± eÙ. Proof. Let 5À J\ q» K be the change of context epimorphism. Since ØeÙnor Ÿ kerÐ5Ñ, we have 8   ÐJ\ À ØeÙnor Ñ   ÐJ\ À kerÐ5ÑÑ œ kK k œ 8

from which it follows that kerÐ5Ñ œ ØeÙnor and so K¸

J\ ØeÙnor

…

Referring to our previous example, since the dihedral group H) fits description 2) and has order ), we have H) ¸ Ø+ß , ± +# œ "ß , % œ "ß +,+, œ "Ù More generally, one of the simplest presentations with two generators is L œ ØBß C ± B8 œ "ß C7 œ "ß CB œ BC > Ù for some !  >  7. The commutativity relation shows that L œ ÖB3 C4 ± ! Ÿ 3  8 and ! Ÿ 4  7× and so 9ÐLÑ Ÿ 78. Moreover, one can prove by induction that for ! Ÿ 5  7 and ! Ÿ 4  8, 4

C5 B4 œ B4 C5> and so

4

ÐB3 C5 ÑÐB4 Cj Ñ œ B34 C5> j

(12.28)

However, we can define a group K whose underlying set consists of the 78 distinct formal symbols

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Fundamentals of Group Theory

K œ ÖB3 C4 ± ! Ÿ 3  8 and ! Ÿ 4  7× with product defined by (12.28). This product is associative, since 4

4

?

ÒÐB3 C5 ÑÐB4 Cj ÑÓÐB? C @ Ñ œ ÐB34 C 5> j ÑÐB? C @ Ñ œ B34? C Ð5> jÑ>

@

and ÐB3 C5 ÑÒÐB4 Cj ÑÐB? C@ ÑÓ œ B3 C5 ÐB4? Cj>

?

@

Ñ œ B34? C5>

4?

j>?@

and inverses exist, since ÐB3 C5 Ñ" œ B83 C75>

83

Hence, K is a group of size 87 that satisfies e and so K has presentation Ø\ ± eÙ and 9ÐLÑ œ 78. Theorem 12.29 1) The presentation Ø\ ± eÙ œ Ø+ß , ± +8 œ "ß , 7 œ "ß ,+ œ +, >Ù where !  >  7 defines the group K œ Ö+3 , 4 ± ! Ÿ 3  8 and ! Ÿ 4  7× where 9ÐKÑ œ 78, 9Ð+Ñ œ 8, 9Ð,Ñ œ 7 and 4

Ð+3 , 5 ÑÐ+4 , j Ñ œ +34 , 5> j Moreover, any group of order 78 that is generated by \ and satisfies the relations e has presentation Ø\ ± eÙ. 2) The presentation Ø] ± f Ù œ Ø-ß . ± - 8 œ "ß . 7 œ "ß .- œ - = .Ù defines the group L œ Ö- 3 . 4 ± ! Ÿ 3  8 and ! Ÿ 4  7× where 9ÐLÑ œ 78, 9Ð-Ñ œ 8, 9Ð.Ñ œ 7 and 5

Ð- 3 . 5 ÑÐ- 4 . j Ñ œ - 34= . 5j Moreover, any group of order 78 that is generated by ] and satisfies the relations f is defined by Ø] ± f Ù.… Let us now consider some examples of presentations.

Free Groups and Presentations

347

Dihedral Groups Since the dihedral group H#8 œ Ø5ß 3Ù of order #8 satisfies the relations e œ Ö5# œ "ß 38 œ "ß 35 œ 538" × Theorem 12.29 implies that H#8 multiplication table

has presentation ØÖ5 ß 3× ± eÙ and 4

Ð53 35 ÑÐ54 3j Ñ œ 534 35Ð8"Ñ j We leave it as an exercise to show that H#8 is also presented by Ø] ± f Ù œ ØBß C ± B# œ "ß C# œ "ß ÐBCÑ8 œ "Ù Thus, two rather different looking presentations can be equivalent, that is, can present the same group.

Quaternion Group To find a presentation for the quaternion group, note that U œ Ø3ß 4Ù satisfies the relations e œ Ö3% œ "ß 3# œ 4# ß 43 œ 3$ 4× and so we need only show that any group presented by Ø\ ± eÙ has order at most ). If K has presentation Ø\ ± eÙ œ ØBß C ± B% œ "ß C# œ B# ß CB œ B$ CÙ then K œ ÖB= C> ± ! Ÿ =ß > Ÿ $× However, since C# œ B# , we see that K œ ÖB= C> ± ! Ÿ = Ÿ $ß ! Ÿ > Ÿ "× and so 9ÐKÑ Ÿ ). Thus, U ¸ Ø\ ± eÙ.

Dicyclic Groups Consider the presentation Ø\ ± eÙ œ ØBß C ± B#8 œ "ß C# œ B8 ß CB œ B" CÙ If L œ Ø\ ± eÙ then using the fact that C# œ B8 , we have

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Fundamentals of Group Theory

L œ ÖB3 C4 ± ! Ÿ 3 Ÿ #8  "ß ! Ÿ 4 Ÿ "× and so 9ÐLÑ Ÿ %8. A double induction shows that for ! Ÿ 5  % and ! Ÿ 4  #8, C5 B4 œ BÐ"Ñ 4 C5 œ œ 5

B4 C 5 B#84 C5

5 even 5 odd

and so 5

ÐB3 C5 ÑÐB4 Cj Ñ œ B3Ð"Ñ 4 C5j

(12.30)

However, we can define a group K by choosing two distinct symbols B and C and setting K œ ÖB3 C4 ± ! Ÿ 3  #8ß 4 œ !ß "× with product defined by (12.30). This product is associative: 5

5

ÒÐB3 C5 ÑÐB4 Cj ÑÓÐB? C@ Ñ œ ÐB3Ð"Ñ 4 C5j ÑÐB? C@ Ñ œ B3Ð"Ñ

4Ð"Ñ5j ? 5j@

C

and j

ÐB3 C5 ÑÒÐB4 Cj ÑÐB? C@ ÑÓ œ B3 C5 ÐB4Ð"Ñ ? Cj@ Ñ œ B3Ð"Ñ

5

Ð4Ð"Ñj ?Ñ 5j@

C

and inverses exist: 5

ÐB3 C5 Ñ" œ BÐ"Ñ

Ð#83Ñ %5

C

Hence, K is a group of size %8 that satisfies the relations e and so K ¸ Ø\ ± eÙ and 9ÐLÑ œ %8. Any group with presentation Ø\ ± eÙ is called a dicyclic group of order %8. A special case of the dicyclic group is when #8 has the form #8" , in which case the presentation is 8"

Ø\ ± eÙ œ ØBß C ± B#

8#

œ "ß C # œ B#

ß CB œ B" CÙ

A group with this presentation is called a generalized quaternion group. When 8 œ $, this is Ø\ ± eÙ œ ØBß C ± B% œ "ß C# œ B# ß CB œ B" CÙ which is the presentation for the quaternion group U.

The Symmetric Group Recall from Theorem 6.5 that the symmetric group W8 is generated by the 8  " adjacent transpositions >5 œ Ð5 5  "Ñ for 5 œ "ß á ß 8  ". Note also that the >5 's satisfy the relations

Free Groups and Presentations

>#5 œ "ß

5 >>55" œ >>5" ß

349

>5 >4 œ >4 >5 for 4  5 Á „"

Now let \ œ ÖB" ß á ß B8" × and let e be the relations B#5 œ "ß

B5" B5 B5" œ B5 B5" B5 ß

B5 B4 œ B4 B5 for 4  5 Á „"

Let K œ Ø\ ± eÙ. Since W8 is generated by the elements >5 œ Ð5 5  "Ñ and satisfies the relations e with B5 replaced by >5 , Theorem 12.27 implies that if 9ÐKÑ Ÿ 8x, then W8 ¸ Ø\ ± eÙ. We prove the former by induction on 8. If 8 œ #, then K œ ØB" Ù where B#" œ " and so K œ Ö"ß B" × has order #. Assume that the result holds for the subgroup L œ ØB" ß á ß B8# Ù with relations e. We show that every +  L can be written in the form + œ B5 B5" âB8" A for " Ÿ 5 Ÿ 8  ", where A − L . Let us refer to a substring B3 B3" âB8" as being in proper order. Then + œ ?ÐB3 B3" âB8" Ñ@ for some 3 Ÿ 8  ", where @ − L . If ? is the empty string, then we are done. Otherwise, let ? œ ?w B4 . Here are the possibilities: 1) If 4 Ÿ 3  #, then B4 commutes with all factors to its right and so + œ ?w ÐB3 B3" âB8" ÑB4 @ where B4 @ − L . 2) If 4 œ 3  ", then + œ ?w ÐB3" B3 B3" âB8" Ñ@ 3) If 4 œ 3, then + œ ?w B3 ÐB3 B3" âB8" Ñ@ œ ?w ÐB3" âB8" Ñ@ 4) If 4   3  ", then + œ ?w B4 ÐB3 B3" âB8" Ñ@ œ ?w ÐB3 âB4 B4" B4 âB8" Ñ@ œ ?w ÐB3 âB4" B4 B4" âB8" Ñ@ œ ?w ÐB3 âB4" B4 âB8" ÑB4" @ where B4" @ − L .

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Thus, in all cases, we can reduce the length of the substring appearing to the left of the substring in proper order by one symbol. Repeated application brings + to the desired form + œ B5 B5" âB8" A It follows that 9ÐKÑ œ 8 † 9ÐLÑ Ÿ 8x. Theorem 12.31 The symmetric group W8 has presentation Ø\ ± eÙ, where \ œ ÖB" ß á ß B8" × and let e consist of the relations B#5 œ "ß

B5" B5 B5" œ B5 B5" B5 ß

B5 B4 œ B4 B5 for 4  5 Á „"

…

We close by noting that the relations above are equivalent to B#5 œ "ß

ÐB5" B5 Ñ$ œ "ß

ÐB5 B4 Ñ# œ " for 4 Ÿ 5  #

Exercises 1. 2.

An equational class ^ is abelian if all members of the class are abelian groups. Characterize abelian equational classes. Let K be the dicyclic group of order %8, 8  ", with presentation Ø\ ± eÙ œ ØBß C ± B#8 œ "ß C# œ B8 ß CB œ B#8" C œ B" CÙ

3.

a) K has exactly one involution D . b) ^ÐKÑ œ ØDÙ c) KÎ^ÐKÑ ¸ H#8 Let \ be a nonempty set and let ÐJ\ ß , Ñ be universal for \ . Let AÐB" ß á ß B8 Ñ be a word over \ w and let 0 À ÖB" ß á ß B8 × Ä ÖC" ß á ß C8× be an injection, where C3 − \ . Prove that AÐ,B" ß á ß ,B8 Ñ œ "

4. 5.

6.

7. 8.

Í

AÐ,C" ß á ß ,C8 Ñ œ "

Let J be the free group on the set \ œ ÖB" ß á ß B8 ×. Show that J has a subgroup of index 7 for all " Ÿ 7 Ÿ 8. Let J be the free group on \ œ ÖBß C× and let K œ Ø+Ù } Ø,Ù be the direct product of two infinite cyclic groups. The function 0 À \ Ä K defined by 0 B œ Ð+ß "Ñß 0 C œ Ð"ß ,Ñ induces a unique mediating morphism 7 À J Ä K . What is the kernel of 7 ? Let J\ be the free group on \ . Prove the following: a) If ] © \ is nonempty, then J] Ÿ J\ . b) If ] © \ is nonempty, then J] ∩ J\Ï] œ Ö"×. In particular, if B − \ Ï ] , then B  J] . Characterize the abelian groups that are free groups. Let J be a free group and let L Ÿ J have finite index. Show that L intersects every nontrivial subgroup of J nontrivially.

Free Groups and Presentations 9. 10. 11. 12. 13.

14.

351

Let J be free on the disjoint union \ ∪ ] of nonempty sets. Prove that J ÎØ] Ùnor is free on \ . Show that if R ü K and KÎR is free, then K œ R z L for some L Ÿ K. Prove that if k\ k  ", then the free group J\ is centerless. Let J be a free group. Suppose that 5À E q» F is an epimorphism and that 7 À J Ä F is a homomorphism. Prove that there is a -À J Ä E for which 5 ‰ - œ 7 . This is called the projective property of free groups. If ^ is a class of groups, then a group K is residually ^ or a residually ^group if for any " Á + − K, there is a normal subgroup R+ ü K for which +  R+ and KÎR+ is a ^-group. a) Prove that a group K is residually ^ if and only if it is isomorphic to a subdirect product of ^-groups. b) Prove that if a œ ÖR3 ± 3 − M× is a family of normal subgroups of a group K and if KÎR3 is a ^-group for all 3 − M , then KÎ+R3 is residually ^. Prove that the presentation Ø] ± f Ù œ Ø-ß . ± - 8 œ "ß . 7 œ "ß .- œ - = .Ù defines the group L œ Ö- 3 . 4 ± ! Ÿ 3  8 and ! Ÿ 4  7× where 9ÐLÑ œ 78, 9Ð-Ñ œ 8, 9Ð.Ñ œ 7 and 5

Ð- 3 . 5 ÑÐ- 4 . j Ñ œ - 34= . 5j 15. Show that H#8 is presented by Ø] ± f Ù œ ØBß C ± B# œ "ß C# œ "ß ÐBCÑ8 œ "Ù 16. Let 5 and 3 be distinct symbols. Let H œ Ö33 ß 533 ± 3 − ™× with product defined by the properties 5# œ "ß 35 œ 53" . Thus, 33 5 œ 533 a)

Show that H is a group and that H is presented by T" œ ØBß C ± B# œ "ß CB œ BC" Ù

b) Show that H is also presented by T# œ ØBß C ± B# œ "ß C# œ "Ù Any group presented by T" or T# is called an infinite dihedral group. 17. Let K œ Ø\ ± eÙ be a finite presentation of K, where \ œ ÖB" ß á ß B8 × and ew œ Ö<" ß á ß <7 × and 7  8. Show that K is an infinite group as follows.

352

Fundamentals of Group Theory a)

Reduce the problem to the abelian case as follows. Let E\ be the free abelian group on \ . Show that there is an epimorphism from J\ ÎØew Ùnor to E\ ÎØew Ù. b) Show that E\ ÎØew Ù is infinite.

Chapter 13

Abelian Groups

In this chapter, we study abelian groups. We will write abelian groups using additive notation. One of our main goals is to provide a complete solution to the classification problem for finitely-generated abelian groups. That is, we will describe all finitely-generated abelian groups up to isomorphism. Perhaps the most natural place to begin is to observe that the elements of finite order in an abelian group E form a subgroup of E, since 9Ð+,Ñ œ lcmÐ9Ð+Ñß 9Ð,ÑÑ Let us remind the reader that this is not the case in a general group. For example, in KPÐ#ß ‚Ñ, let EœŒ

! "

" !

! and F œ Œ "

" "

Then E and F have finite order but EF has infinite order. Definition Let E be an abelian group. An element + − E that has finite order is called a torsion element. The subgroup Etor of all torsion elements in E is called the torsion subgroup of E. A group that has no nonzero torsion elements is said to be torsion free and a group all of whose elements are torsion elements is said to be torsion.… Of course, a finite group is a torsion group, but the converse is not true: Consider the direct product ™i# ! of an infinite number of copies of ™# . The quotient EÎEtor is easily seen to be torsion free and it would be nice if Etor was always a direct summand of E, that is, if E œ Etor  F for some Fß E, since then F ¸ EÎEtor would be torsion free and we would

S. Roman, Fundamentals of Group Theory: An Advanced Approach, DOI 10.1007/978-0-8176-8301-6_13, © Springer Science+Business Media, LLC 2012

353

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Fundamentals of Group Theory

have a nice decomposition of any abelian group. Unfortunately, this is not the case. To show this, we require a definition. Definition Let E be an abelian group. An element + − E is divisible by an integer 8 if there is an element , − E for which + œ 8, . A group E is divisible if every element is divisible by every nonzero integer.… Theorem 13.1 The torsion subgroup of an abelian group need not be complemented. Proof. Let E œ } ™: be the external direct product of the abelian groups ™: , taken over all primes :. For + − E, we use the notation +: in place of +Ð:Ñ. The torsion subgroup Etor is the subgroup of all elements with finite support. If E œ Etor  F , then F ¸ EÎEtor , so this prompts us to look for an isomorphism-invariant property that holds in EÎEtor but not in F . This property is divisibility. Specifically, we will show the following: 1) No nonzero element of E (and hence F ) is divisible by all primes :. 2) There are nonzero elements of EÎEtor that are divisible by all primes :. Since the only element of ™: that is divisible by : is !, if + − E is divisible by :, then +: œ !. Hence, if + is divisible by all primes, it follows that + œ !. Now let + − E be the element for which +: œ " for all : . For a given prime : , to say that + œ :, for some , − E is to say that :,; œ " for all primes ; . But if ; Á :, then : has an inverse <: in the field ™; . Hence, if , is defined by ,: œ <: and ,: œ !, then Ð+  :,Ñ: œ +:  :,: œ " and for all ; Á :, Ð+  :,Ñ; œ +;  :,; œ ! and so +  :, − Etor .… Despite the negative nature of the previous result, we will show that if E is a finitely-generated abelian group, then Etor is complemented. This is a key to the structure theorem for finitely-generated abelian groups.

Abelian Groups

355

An Abelian Group as a ™-Module An abelian group E has a natural scalar multiplication defined upon it, namely, multiplication by the integers: If α − ™ and + − E, we set Ú Ý Ý! +â+ α+ œ Û ðóóñóóò α terms Ý Ý Ü ÐαÑ+

if α œ ! if α  !

(13.2)

if α  !

Under this operation, an abelian group E is a ™-module, as defined below. Definition Let V be a commutative ring with identity. An V-module (or a module over V ) is an abelian group Q , together with a scalar multiplication, denoted by juxtaposition, that assigns to each pair Ð<ß ?Ñ − V ‚ Q , an element <@ − Q . Furthermore, the following properties must hold for all <ß = − V and ?ß @ − Q : <Ð?  @Ñ œ
The Classification of Finitely-Generated Abelian Groups We solved the classification problem for finite abelian groups in Theorem 5.7. Also, Theorem 12.14 solves the classification problem for free abelian groups. Using these theorems, we can now solve the classification problem for finitelygenerated abelian groups. The first step is to note the following. Theorem 13.3 A finitely-generated abelian group E is torsion free if and only if it is free. Proof. We leave proof that if E is free, then it is torsion free as an exercise. For the converse, let W œ Ö@" ß á ß @8 × be a generating set for the torsion-free abelian group E. The proof is based on the fact that since E is torsion free, it is a torsion-free ™-module. Moreover, for any + − E, the multiplication map .+ À E Ä E defined by .+ B œ +B is a ™-module automorphism of E.

356

Fundamentals of Group Theory

Let Ö?" ß á ß ?5 × be a maximal linearly independent subset of W . Of course, if 5 œ 8, then W is a basis for E and so E is free. Assume otherwise and let W œ Ö?" ß á ß ?5 ß @" ß á ß @85 × For each @3 , the set Ö?" ß á ß ?5 ß @3 × is linearly dependent and so there exist +3 − ™ for which +3 @3 − Ø?" ß á ß ?5 Ù If + œ α" âα85 , then .+ E œ +Ø?" ß á ß ?5 ß @" ß á ß @85 Ù © Ø?" ß á ß ?5 Ù and since the latter is a free abelian group, Theorem 12.18 implies that .+ E is also free and therefore so is E.… If E is a finitely-generated abelian group E, then Etor is a subgroup of E and the quotient EÎEtor is torsion-free and finitely generated and so is free. Since the canonical projection map 1À E Ä EÎEtor is an epimorphism, Theorem 12.19 implies that Etor has a complement: E œ J  Etor where J ¸ EÎEtor is free and finitely-generated. Moreover, Theorem 12.16 implies that J has finite rank and since Etor is finitely generated (Theorem 2.21), torsion and abelian, it is finite. As to uniqueness of this decomposition, if E œ Jw  X where J w is free and X is torsion, then clearly, X Ÿ Etor . But if + − Etor and + œ >  0 where > − X and 0 − J w and so 0 œ +  > is torsion, whence 0 œ ! and + − X . Thus, X œ Etor . It follows that J and J w are both complements of Etor and hence are isomorphic. We can now state the fundamental theorem of finitely-generated abelian groups. Theorem 13.4 (The fundamental theorem of finitely-generated abelian groups) Let E be a finitely-generated abelian group, with torsion subgroup Etor . 1) Then E œ J  Etor where J is free of finite rank < and Etor is finite. As to uniqueness, if E œ Jw  X

Abelian Groups

357

where J w is free and X is torsion, then X œ Etor and rkÐJ Ñ œ rkÐJ w Ñ. The number < œ rkÐJ Ñ is called the free rank of E. 2) (Invariant factor decomposition) E is the direct sum of a finite number of cyclic subgroups E œ ØB" Ù  â  ØB< Ù  Ø?" Ù  â  Ø?8 Ù where 9ÐB3 Ñ œ ∞ and 9Ð?3 Ñ œ α3   # and α8 ± α8" ± â ± α" The orders α3 are called the invariant factors of E. 3) (Primary cyclic decomposition) If /

/

α5 œ :"5ß" â:75ß7 then E œ ØB" Ù  â  ØB< Ù  ÒØ?"ß" Ù  â  Ø?"ß5" ÙÓ  â  ÒØ?7ß" Ù  â  Ø?7ß57 ÙÓ /

where 9ÐB3 Ñ œ ∞ and 9Ð?3ß4 Ñ œ :3 3ß4 and /3ß"   /3ß#   â   /3ß53   " /

The numbers :3 3ß4 are called the elementary divisors of E. / 3) The multiset Öα3 × of invariant factors and the multiset Ö:3 3ß4 × of elementary divisors are uniquely determined by the group E.…

Projectivity and the Right-Inverse Property A diagram of the form 5

7

EÒFÒG where E, F and G are groups and 5 and 7 are group homomorphisms is exact if imÐ5Ñ œ kerÐ7 Ñ It is customary to regard the figure 5

EÒFÒ! as exact and to omit the second homomorphism, since it must be the zero map. Thus, this figure says precisely that 5 is surjective. Similarly, the figure 5

!ÒEÒF says precisely that 5 is injective. According to Theorem 5.23, a group homomorphism 5À K Ä Kw has a right inverse 5V if and only if it is surjective and kerÐ5 Ñ is complemented:

358

Fundamentals of Group Theory

K œ kerÐ5 Ñ z O for some O Ÿ K and in this case, K œ kerÐ5Ñ z imÐ5V Ñ Let us say that an abelian group T has the right-inverse property if every epimorphism 5À E q» T , where E is abelian, has a right inverse. This is illustrated in Figure 13.1.

σR A

σ

P ι 0

P

Figure 13.1 An apparently stronger property is given in the following definition.

λ A

σ

P τ B

0

Figure 13.2 Definition An abelian group T is projective if, referring to Figure 13.2, for any epimorphism 5À E q» F of abelian groups and any homomorphism 7 À T Ä F , there is a homomorphism -À T Ä E for which 5‰- œ7 In this case, we say that 7 can be range-lifted to -.… While the projective property appears to be stronger than the right-inverse property, the two properties are actually equivalent. The following theorem is the main result on projective abelian groups. Theorem 13.5 Let K be an abelian group. The following are equivalent: 1) K is a free abelian group. 2) K is projective. 3) K has the right-inverse property, that is, any surjection 7 À E Ä K, where E is abelian, has a right inverse. 4) If 5À E q» K, where E is abelian, then kerÐ5 Ñ is a direct summand of E. Proof. We have seen that 3) and 4) are equivalent. Assume that 1) holds and let K œ J\ be free on \ . Let 5À E Ä F be surjective and let 7 À J\ Ä F . Then for

Abelian Groups

359

each B − \ , there is an +B − E for which 5+B œ 7 B. Define a function 0 À \ Ä E by 0 ÐBÑ œ +B . Since J\ is free, there is a unique homomorphism -À J\ Ä E for which -B œ +B . Then 5 ‰ - ÐBÑ œ 5 +B œ 7 B and so 5 ‰ - œ 7 on \ and therefore on J\ . Hence, K œ J\ is projective and 2) holds. It is clear that 2) implies 3). Finally, suppose that 3) holds. The identity map +À K Ä K can be lifted to a homomorphism 5À JK Ä K where JK is the free abelian group with basis K . Of course, 5 is surjective and so 3) implies that 5 has a right inverse 5V À K Ä JK . Hence, kerÐ5Ñ is complemented, that is, JK œ kerÐ5Ñ  W But 5À W Ä K is an isomorphism and so K is isomorphic to a direct summand of a free abelian group and is therefore free abelian by Theorem 12.18. Hence, 1) holds.…

Injectivity and the Left-Inverse Property We have seen that a monomorphism 5À E ä F has a left-inverse 5P if and only if imÐ5Ñ has a complement O in F , in which case O ¸ kerÐ5P Ñ. Dual to the right-inverse property is the left-inverse property: An abelian group I has the left-inverse property if every monomorphism 5À I ä F to an abelian group F has a left inverse. Dual to the projective property is the injective property. Definition An abelian group I is injective if, referring to Figure 13.3, for any embedding 5À E ä F and any homomorphism 7 À E Ä I there is a homomorphism -À E Ä I for which -‰5 œ7 In this case, we say that 7 À E Ä I can be domain-lifted to -À F Ä I by 5À E Ä F.…

0

A τ

σ

B λ

E Figure 13.3 Baer [3] proved that if this condition holds in the special case where F œ ™ is the group of integers and E Ÿ ™ and 5À E Ä ™ is the inclusion map, then the condition holds in general and I is injective.

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Fundamentals of Group Theory

Theorem 13.6 An abelian group I is injective if and only if it satisfies Baer's criterion: Any homomorphism 7À M Ä I , where M is a subgroup of the integers ™ can be extended to a homomorphism -À ™ Ä I on ™. Proof. We wish to show that any homomorphism 7À E Ä I can be domainlifted to -À F Ä I by any monomorphism 5À E ä F . Note that 7 can be domain-lifted by 5 to 7 ‰ 5" À 5E Ä I , since 5À E Ä 5E is an isomorphism. This is shown in Figure 13.4.

0

A

σ

τ

τσ

σ(A) −1

X

B

λ

E Figure 13.4 Suppose we have domain-lifted 7 by 5 to -À \ Ä F , where 5E Ÿ \ Ÿ F , that is, 7 œ-‰5 If \  F , then for any + − \ Ï F , we can lift - by 5 to a map on \  Ø+Ù as follows. We need only define - on Ø+Ù œ ™+. But - is already defined on ™+ ∩ \ œ M+ for some M Ÿ ^ . So if -" À M Ä I is defined by -" ÐαÑ œ -Ðα+Ñ then Baer's criterion implies that -" can be extended to -# À ™ Ä I . Then the map -À \  Ø+Ù Ä I defined by -ÐB  <+Ñ œ -ÐBÑ  -# Ð<Ñ+ for any < − ™ is well defined since if B  <+ œ C  =+, then B  C œ Ð=  <Ñ+ and so =  < − M , which implies that -" Ð=  <Ñ œ -ÐÐ=  <Ñ+Ñ œ -ÐB  CÑ and so -ÐB  <+Ñ œ -ÐBÑ  -# Ð<Ñ+ œ -ÐCÑ  -# Ð=Ñ+ œ -ÐC  =+Ñ Moreover, since - ‰ 5 œ 7 and imÐ5Ñ Ÿ domÐ-Ñ, it follows that - ‰ 5 œ 7 . This discussion prompts us to apply Zorn's lemma. Let f be the collection of all pairs Ð\ß -Ñ, where 5ÐEÑ © \ © F and - is a lifting of 7 by 5 to a map on \ . Then f is nonempty since we may take \ œ 5E. Order f by setting Ð\ß .Ñ Ÿ Ð] ß -Ñ if \ © ] and -l] œ ..

Abelian Groups

361

If V œ ÖÐ\3 ß .3 Ñ× is a chain in f , let Y œ -\3 . If B − \3 ∩ \4 , then one of .3 and .4 is an extension of the other and so .3 B œ .4 B. Hence, we may define . by .B œ .3 B for any 3 satisfying B − \3 . Then ÐY ß .Ñ is an upper bound for V and so Zorn's lemma implies that f has a maximal element ÐQ ß -Ñ. But if Q  F, then there is a further lifting of -, which contradicts the maximality of ÐQ ß -Ñ and so Q œ F .… We can now present our main theorem on injective groups. Theorem 13.7 Let K be an abelian group. The following are equivalent: 1) K is injective. 2) K is divisible. 3) K has the left-inverse property, that is, every monomorphism 5 À K ä F to an abelian group F has a left inverse. 4) If 5À K ä E, where E is abelian, then imÐ5Ñ is a direct summand of E. Proof. We know that 3) and 4) are equivalent. Assume first that K is injective. For 1 − K and 8 − ™ we seek 2 − K for which 1 œ 82 . The map 7 À Ø8Ù Ä I defined by 7 Ð<8Ñ œ <1 for all < − ™ can be domain-lifted by the inclusion map 4À Ø8Ù Ä ™, that is, -‰4œ7 Hence, 1 œ 7 Ð8Ñ œ - ‰ 4Ð8Ñ œ -Ð8Ñ œ 8-Ð"Ñ and so 2 œ -Ð"Ñ. Thus, K is divisible and 1) implies 2). To see that 2) implies 1), we show that 2) implies the Baer criterion. Let 7 À Ø5Ù Ä K where Ø5Ù Ÿ ™. To show that 7 can be extended to ™, define -+ À ™ Ä K by -+ Ð8Ñ œ 8+, for + − K. Then -+ extends 7 if -+ Ð5Ñ œ 7 Ð5Ñ, that is, if 7Ð5Ñ œ 5+. But since K is divisible, there is an + − K for which this holds. Hence, 2) implies 1). It is clear that 1) implies 3). Finally, suppose that I has the left-inverse property. We wish to show that I is injective. The story of the proof is shown in Figure 13.5.

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Fundamentals of Group Theory

σ

A

0

B κB

λ

τ G

κG

µB

G ûú B

µG

πS G ûú B

S Figure 13.5 We seek the map -À F Ä K for which - ‰ 5 œ 7 . A first attempt might be to consider the direct sum K  F , with canonical injections ,K and ,F . Since ,K À K Ä K  F is an injection, it has a left inverse Ð,K ÑP and we can take - œ Ð,K ÑP ‰ ,F . However, it may not be the case that - ‰ 5 œ 7 . On the other hand, perhaps we can factor the direct sum K  F by a subgroup W, with projection map 1W in such a way that the compositions .F œ 1W ‰ ,F À F Ä ÐK  FÑÎW and .K œ 1W ‰ ,K À K Ä ÐK  FÑÎW satisfy .F ‰ 5 œ .K ‰ 7 where .K is left-invertible. In this case, if - œ Ð.K ÑP ‰ .F À F Ä K , then - ‰ , œ Ð.K ÑP ‰ .F ‰ 5 œ Ð.K ÑP ‰ .K ‰ 7 œ 7 as desired. But the condition .F ‰ 5 œ .K ‰ 7 is 1W ‰ ,F ‰ 5 œ 1W ‰ ,K ‰ 7 that is, Ð!ß 5+Ñ  W œ Ð7 +ß !Ñ  W for all + − E and so if W œ ÖÐ7 +ß 5+Ñ ± + − E× then this equation holds. Also, .K is injective since 1 − kerÐ.K Ñ implies that Ð1ß !Ñ − W and so Ð1ß !Ñ œ Ð7 +ß 5+Ñ. Hence, 5+ œ ! and so + œ !, whence 1 œ !.…

Abelian Groups

363

Exercises 1. 2.

Let E be a torsion-free abelian group. Prove that if + − E is divisible by 8 − ™, then the “quotient” , − E is unique. Prove that a subset U of an abelian group E is a basis if and only if for every + − E, there are unique elements ," ß á ß ,8 − U and unique scalars α" ß á ß α8 − ™ for which + œ α" , "  â  α 8 , 8

3. 4. 5. 6.

Let E be a free abelian group. Show that it is not necessarily true that any linearly independent set of size rkÐEÑ is a basis for E. Let E be a free abelian group of finite rank and let E œ F  G . Prove that rkÐEÑ œ rkÐFÑ  rkÐGÑ. Prove that any abelian group E is isomorphic to a quotient of a free abelian group. A subgroup L Ÿ K of an additive abelian group K is pure if for any 2 − L and 7 − ™, 2 œ 71 for 1 − K

Ê

2 œ 72 w for some 2 w − L

a)

7. 8.

9. 10. 11. 12. 13.

Prove that if K is an abelian group, then the set of periodic elements of K is pure. b) Prove that any direct summand of an abelian group is pure. c) Find a nonpure subgroup of the cyclic group ™8# . Prove that an abelian group E is finitely generated if and only if it is isomorphic to a quotient of a free abelian group of finite rank. Let E be a free abelian group of finite rank 8. Let W œ Ö=" ß á ß =8 × be a generating set for E. Prove that W is a basis for E. Hint: Let \ œ ÖB" ß á ß B8 × be a basis for E and define the map 7 À E Ä E by 7ÐB3 Ñ œ =3 and extending to a surjective homomorphism. What about kerÐ7 Ñ? Let J be a free abelian group of rank 8. Let L be a subgroup of J of rank 5  8. Prove that KÎL contains an element of infinite order. Show that, in general, a basis for a subgroup of a free abelian group cannot be extended to a basis for the entire group. Prove that any free abelian group is torsion free. Let K be a torsion-free abelian group. Suppose that K has a subgroup J that is free and has finite index. Prove that K is free abelian. Let E be a finite abelian group. a) Prove that if :E œ Ö!×, then E is a vector space over the field ™: œ ™Î:™. b) Prove that for any subgroup W of E the set W Ð:Ñ œ Ö@ − W ± :@ œ !×

364

Fundamentals of Group Theory is also a subgroup of E and if E œ W  X , then EÐ:Ñ œ W Ð:Ñ  X Ð:Ñ

14. Let J be a free abelian group of rank 8. Show that AutÐJ Ñ is isomorphic to the group of all 8 ‚ 8 matrices with determinant equal to „". 15. Let  be the multiplicative group of positive rational numbers. a) Show that  is isomorphic to the additive group ™ÒBÓ of polynomials over the integers. Hint: Use the fundamental theorem of arithmetic. b) Show that the multiplicative group  of positive rationals is a free abelian group of countably infinite rank. 16. Prove that the quasicyclic group ™Ð:∞ Ñ is divisible. 17. Prove that a free abelian group is not divisible. 18. a) Show that the rational numbers  are not finitely generated as an abelian group under addition. b) Show that  is divisible and therefore not free by an earlier exercise. c) Show that  is torsion free. Thus, a torsion-free abelian group need not be free. 19. Prove that if E œ } E3 , where E and E3 are abelian groups, then E is injective if and only if E3 is injective for all 3. 20. Let E be an abelian group. Show that the set Ediv of all divisible elements is a subgroup of E. Show that Ediv is a direct summand of E. 21. Let H be a divisible group. Prove that Htor is divisible. 22. Let E be a finitely generated abelian group. For any subgroup W of E let EÐ:Ñ œ Ö+ − E ± + is a :-element× Show that EÐ:Ñ « E. Describe the order of EÐ:Ñ in terms of the elementary divisors of E. 23. How can one tell from the elementary divisors of a finite abelian group when that group is cyclic? 24. Use one of the decomposition theorems to prove that a finite abelian group E has a subgroup of order 5 for every 5 ± 9ÐEÑ. 25. Find, up to isomorphism, all finite abelian groups of order "!!!. Which are cyclic? 26. Prove that every abelian group of order 426 is cyclic. 27. Let : be a prime. Let E œ Ø?" Ù  â  Ø?7 Ù where 9Ð?3 Ñ œ :03 and let F œ Ø@" Ù  â  Ø@8 Ù where 9Ð@3 Ñ œ :/3 . Assume that E Ÿ F . a) Prove that 7 Ÿ 8.

Abelian Groups b) Prove that 07 Ÿ /8 ß 07" Ÿ /8" ß á ß 0" Ÿ /87"

365

References

[1] [2] [3] [4] [5] [6]

[7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

Allen, W., Complex Groups, The American Mathematical Monthly, Vol. 67, No. 7 (1960) 637–641. Aschbacher, M., The Status of the Classification of Finite Simple Groups, Notices of the AMS, Volume 51, Number 7, 2004. Baer, R., Situation der Untergruppen und Struktur der Gruppe, S.-B. Heidelberg. Akad. Math.-Nat. Klasse 2 (1933) 12–17. Brauer, R. and Fowler, K. A., On Groups of Even Order, Annals of Mathematics, 2nd Ser., Vol. 62, No. 3 (1955) 565–583. Cassidy, P. J., Products of commutators are not always commutators: An example, The American Mathematical Monthly, Vol. 86, No. 9 (1979) 772. Cauchy, A.-L., Mémoire sur les arrangements que l’on peut former avec des lettres données et sur les permutations ou substitutions à l’aide desquelles on passe d’un arrangement à un autre. Exercises d’analyse et de physique mathématique 3 (1845) 151–252. (Reprinted in: Oeuvres complètes, vol. 13, second series. Gauthier-Villars, Paris (1932) pp. 171– 282.) Cayley, A., On the theory of groups as depending on the symbolic equation ) 8 œ ", Phil. Mag., 7 (4) (1854), 40–47. Christensen, C., Complementation in groups, Math. Zeitschr. 84 (1964) 52–69. Christensen, C., Groups with complemented normal subgroups, J. London Math. Soc., 42 (1967) 208–216. Feit, W. and Thompson, J. G., A Solvability Criterion for Finite Groups and Some Consequences, Proc. Nat. Acad. Sci. USA 48 (1962) 968–970. Feit, W. and Thompson, J. G., Solvability of Groups of Odd Order, Pacific J. Math 13 (1963) 775–1029. Frattini, G., Intorno alla generazione dei gruppi di operazioni, Rend. Atti Accad. Lincei, (4) 1 (1885) 281-285, 455–457. Frobenius, G., Berliner Sitzungsberichte, (1895) 995. Fuchs, L., Abelian Groups, Pergammon Press, 1960. Guralnick, R., Expressing group elements as commutators, Rocky Mountain J. Math., Vol. 10, No. 3 (1980), 651–654. Hall, M., The Theory of Groups, AMS Chelsea, 1976.

S. Roman, Fundamentals of Group Theory: An Advanced Approach, DOI 10.1007/978-0-8176-8301-6, © Springer Science+Business Media, LLC 2012

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[17] Hall, P., Complemented groups, J. London Math. Soc., 12, (1937), 201– 204. [18] Hirshon, R., On Cancellation in Groups, American Mathematical Monthly, Vol. 76, No. 9 (1969) 1037–1039. [19] Kappe, L.-C. and Morse, R., On commutators in groups, Proceedings of Groups St. Andrews 2005 LMS Lecture Series, also available online at http://faculty.evansville.edu/rm43/publications/commutatorsurvey.pdf. [20] Kappe, L.-C. and Morse, R., On commutators in p-groups, J. Group Theory, Vol. 8, No. 4 (2005), 415–429. [21] Kertész, A., On groups every subgroup of which is a direct summand, Publ. Math. Debrecen, 2 (1952), 74–75. [22] MacDonald, I. D., Commutators and their products, Amer. Math. Monthly, Vol. 93, No. 6 (1986), 440–444. [23] McKay, J. H., Another proof of Cauchy’s group theorem, American Mathematical Monthly 66, 1959, 119. [24] Meo, M., The mathematical life of Cauchy's group theorem, Historia Mathematica 31 (2004) 196–221. [25] Miller, G. A., The regular substitution groups whose orders is less than 48, Quarterly Journal of Mathematics, 28, (1896), 232–284. [26] Robinson, D., A Course in the Theory of Groups, Springer, 1996. [27] Roman, S., Field Theory, Second Edition, Springer, 2006. [28] Roman, S., Advanced Linear Algebra, Third Edition, Springer, 2007. [29] Schreier, O., Über den Jordan-Hölderschen Satz, Abh. math. Sem. Univ. Hamburg 6 (1928), 300-302. [30] Schmidt, R., Subgroup Lattices of Groups, De Gruyter Expositions in Mathematics 14, 1994. [31] Spiegel, E., Calculating commutators in groups. Math. Mag., Vol. 49, No. 4 (1976), 192–194. [32] Sylow, L., Théorèmes sur les groupes de substitutions, Math. Ann. 5 (1872), 584–594. [33] Weber, H., Lehrbuch der Algebra, Vol 2. Braunschweig, second edition, 1899. [34] Weigold, J., On Direct Factors in Groups, J. London Math Soc., 35 (1960), 310–320. [35] Wielandt, H., Eine Verallgemeinerung der invarianten Untergruppen, Math. Z., 45 (1939), 209–244. [36] Weinstein, M., Examples of Groups, Second Edition, Polygonal Publishing, 2000. [37] Zassenhaus, H. J., Qum Satz von Jordan-Hölder-Schreier, Abh. Math. Semin. Hamb. Univ. 10 (1934) 106–108.

References on the Burnside Problem [38] Adjan, S.I., The Burnside problem and identities in groups, Translated from the Russian by John Lennox and James Wiegold, Ergebnisse der

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Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 95 (1979), Springer-Verlag. [39] Burnside, W., On an Unsettled Question in the Theory of Discontinuous Groups, Quart. J. Pure Appl. Math. 33 (1902) 230–238. [40] Golod, E. S., On Nil-Algebras and Residually Finite -Groups, Isv. Akad. Nauk SSSR Ser. Mat. 28 (1964) 273–276. [41] Hall, M., Solution of the Burnside Problem for Exponent Six, Ill. J. Math. 2 (1958) 764–786. [42] Ivanov, S., On the Burnside Problem on Periodic Groups, Bulletin of the American Mathematical Society, Volume 27, Number 2 (1992) 257–260. [43] Levi, F. and van der Waerden, B. L., Über eine besondere Klasse von Gruppen, Abh. Math. Sem. Univ. Hamburg 9 (1933) 154–158. [44] Novikov, P. S. and Adjan, S. I., Infinite Periodic Groups I, II, III., Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968) 212–244, 251–524, and 709–731. [45] Sanov, I. N., Solution of Burnside's problem for exponent four, Leningrad State Univ. Ann. Math. Ser. 10 (1940) 166–170.

List of Symbols

Ÿ : subgroup ¡ : + ¡ , means that , covers + ³ the item on the left is defined by the item on the right œÀ the item on the right is defined by the item on the left “ : disjoint union of sets ‚ : cartesian product  : internal direct sum; Š in the abelian case }: external direct product {: external direct sum z : semidirect product O z L mod N : O ü Kß K œ LOß L ∩ O œ N « : L « K means that L is characteristic in K ö : L ö K means that L is characteristic and proper in K ä : denotes an embedding (injective map) Ò?Ó: the equivalence class containing ? Ð5ß 8Ñ: If 5 and 8 are integers, this is gcdÐ5ß 8Ñ ACC: ascending chain condition BCC: both chain conditions DCC: descending chain condition WfÐKÑ: the set of direct summands in K G8 Ð+Ñ: the cyclic group Ø+Ù of order 8 ncÐ\ß KÑ: the normal closure of \ in K +: identity map M8 œ Ö"ß á ß 8× ]: for Sylows so we do not conflict with symmetric group. FÐKÑ: the Frattini subgroup of K ÐK À LÑ: index of L in K gcdÐ+ß ,Ñ: greatest common divisor of + and , lcmÐ+ß ,Ñ: least common multiple of + and , ÒLß OÓ: the commutator subgroup of L and O L ‰ : the normal interior of L L ì O : The set product LO where L and O are essentially disjoint. LÐ3Ñ À 1ÖL4 ± 4 Á 3× 371

372

List of Symbols

LÐ5Ñ À a term in the sequence of normal closures of L Ÿ K +Q œ Ö+7 ± 7 − Q × conjK ÐLÑ: the set of conjugates of L by K Fix\ ÐKÑ: the set of elements of \ fixed by the action of K L K : the normal closure of L in K >K ÐLÑ œ ÒLß KÓ 'K ÐLÑ œ \ where \ÎL œ ^ÐKÎLÑ VÒKÓ: the set of commutators of K ^ÐKÑ: the set of all normal subgroups of K that belong to class ^ KÎ^: the set of all normal subgroups R of K for which KÎR belongs to class ^ _Ð^ÑÀ The set of laws of ^ groups subÐKÑ: the lattice of subgroups of a group K sub. ÐKÑ: the subgroups of a finite :-group K of order :. H-subÐKÑ: the lattice of H-subgroups of a group K norÐKÑ: the lattice of normal subgroups of a group K nor. ÐKÑ: the normal subgroups of a finite : -group K of order :. H-norÐKÑ: the lattice of normal H-subgroups of a group K subÐR à KÑÀ the family of all subgroup of K containing R subnH ÐLà KÑ: the family of all H-subnormal subgroups of an H-group K that contain L subnH ÐKÑ: the family of all H-subnormal subgroups of an H-group K Syl: ÐKÑ: the Sylow :-subgroups of K Syl: ÐWà KÑ: the Sylow :-subgroups of K that contain W suppÐ0 Ñ: the support of 0 =ÐLß KÑÀ the length of the sequence of normal closures of L in K f. ÐKÑ: for a :-group K, the set of subgroups of order :. a. ÐKÑ: for a :-group K, the set of normal subgroups of order :. kÐWÑ À the power set of a set W bCompSerH ÐKÑ: K has an H-composition series bCompSerH ÐLà OÑ: K has a composition series from L to O ™‡8 : Ö+ − ™8 ± Ð+ß 8Ñ œ "× SDR: system of distinct representatives homÐKß LÑ: The set of all homomorphisms from K to L › : the restricted wreath product ›› : the complete wreath product ›› < : the regular wreath product ü : normal – : normal and proper ü ü : subnormal – – : subnormal and proper \ w œ \ “ \ " , where \ is a nonempty set

Index

#-transitive, 230 %-group, 24 abelian series, 293 abelian, 20 abnormal, 99, 262, 314 ACC, 6, 60, 76, 136 acceptable, 55 aC-group, 165 action, 123, 207 acts, 123, 207 addition, 20 aD-group, 165 adjacent, 189 affine transformation, 101 algebraic closure, 305 alphabet, 2 alternating group, 33, 203 aNC-group, 165 antichain, 3 antisymmetry, 2 antitone, 5 aperiodic, 21 8: -argument, 245 ascending chain condition, see ACC associativity, 13, 20 asymmetry, 3 automorphism, 106 Baer, 169 Baer's criterion, 360 base field, 305 base ring, 355

base, 181 basis, 319, 326 BCC, 6, 76, 137 Bernstein Theorem, 12 binary operation, 19 Birkhoff, 329, 330 block, 10, 212 both chain condition, see BCC bottom, 4 Brauer, 270 Burnside Theorem, 310 Burnside Basis Theorem, 218 Burnside problem, 35 butterfly lemma, 119 cancellable 7, 157 canonical forms, 10, 11 canonical projection, 108 Cantor's theorem, 12 cardinal number, 12 cardinality, 12 cartesian product, 13 Cassidy, 87 Cauchy's theorem, 79, 124 center, 33, 81 center-intersection property, 215 centerless, 33, 81 central in, 117 central series, 293 central, 33, 81, 144, 216 centralizer, 82 chain, 3 change of context map, 338 characteristic series, 77 373

374

Index

characteristic, 69 characteristically simple, 121 chief distance, 281 chief factors, 278 chief length, 281 chief series, 278 Chinese remainder theorem, 146, 186 class equation, 209, 210 class, 291 ^= -class, 292 ^8 -class, 292 classification problem, 108, 263 closed interval, 3 closed, 19 closure, 138 co-Hopfian, 147 cointersection, 291 cojoin, 292 combinatorial group theory, 343, 344 commutative, 20 commutativity rule, 151 commutativity, 13 commutator subgroup, 84, 146, 307 commutator, 84, 93 commute elementwise, 72 commute, 20, 111 complement, 149, 151 complemented, 149 complements modulo, 172 complete invariant, 11 complete lattice, 8 complete sublattice, 9 complete system of invariants, 11 complete wreath product, 181 complex group, 102 complex product, 31 complexes, 31 component, 73 componentwise product, 24 H-composition distance, 281 composition factors, 278 H-composition factors, 278 H-composition length, 281 composition series, 278

H-composition series, 278 concatenation, 294 concrete free abelian group, 331 concrete free group, 325 concrete -free group, 328 concrete presentation, 339 K-congruence relation, 211 congruence relation, 67 conjugacy class, 30, 82, 84, 205, 214 conjugate, 29 conjugation by, 30 continuum, 14 coordinate, 152 core, 125 correspondence theorem, 113 H-correspondence theorem, 276 coset product rule, 66 coset representative, 42 countable, 12 countably infinite, 12 :-cover, 216 covers, 4 cycle decomposition, 25, 192 cycle representation, 192 cycle structure, 25, 192 cycle, 24, 191 cyclic group, 22 cyclic series, 293 cyclic subgroup, 34 DCC, 6, 76, 137 decidable, 343 decision problem, 343 decision procedure, 343 Dedekind law, 34 Dedekind, 85 defined by generators and relations, 338 derived length, 308 derived series, 308 derived subgroup, 84 descending chain condition, see DCC dicyclic group, 53, 348, 351 direct complement, 151

Index direct factor, 75, 153 direct product, 74, 75, 151, 153 direct sum, 75, 153 direct summand, 75, 153 directed, 3, 37 disjoint union, 13 disjoint, 25, 192 distributive lattice, 60 distributive laws, 60 distributivity, 13 divisible, 145, 147, 354 domain-lifted, 359 double cosets, 233 edges, 189 :-element, 80, 215 elementary abelian group, 121 elementary divisors, 160, 357 embedded, 106 embedding, 106 empty word, 2 endomorphism, 106 endpoint, 77 epimorphism, 105 equational class, 327 equivalence class, 10 equivalence modulo, 41 equivalence relation, 10 equivalence, 230 K-equivalent, 208 H-equivalent, 277 equivalent, 230, 347 essentially disjoint, 33 essentially disjoint product, 33, 151 essentially unique, 286 Euler phi function, 43 Euler's formula, 44 Euler's theorem, 44 even parity, 194 even permutation, 26 even, 194 exact, 357 exponent, 21, 28 extended centralizer, 266 extension problem, 279 extension, 115, 177, 279, 291

375

external direct product, 24, 152 external direct sum, 152 external semidirect product, 176 H-factor group, 274 factor group, 67 factored through, 319 factored uniquely, 111 faithful, 123, 207, 208 Feit, 310 Feit–Thompson, 84 Fermat's little theorem, 44 field extension, 305 finitary operation, 19 finite exponent, 28 finite, 1, 12, 20, 338 finitely f -generated, 138 finitely H-generated, 275 finitely generated, 35 finitely presented, 340 first isomorphism theorem, 112 first H-isomorphism theorem, 276 Fitting's Lemma, 141 Fitting's Theorem, 302 fix, 207 Fowler, 270 Frattini argument, 210 Frattini subgroup, 127 Frattini, 244 free generators, 319, 326 free group, 319 free ^-group, 326 free presentation, 338 free rank, 357 Frobenius, 240 fully invariant, 69 fully-invariant series, 77 fundamental theorem of finitelygenerated abelian groups, 356 K-congruence relation, 211 K-equivalent, 208 K-set, 207 Galois group, 305 Galois-style group, 57

376

Index

Gaussian coefficients, 233 general Burnside problem, 35 general linear group, 23 generalized quaternion group, 222, 348 generalized subnormal join property, 134 generalized symmetric group, 185 8-generated, 35 f-generated, 138 generating set, 34 8-generator group, 35 generators, 338 good order, 162 graph, 189 greatest lower bound, 4 group homomorphism, 46, 105 group with operators, 274 group, 19 ^-group, 291 ^= -class, 292 ^8 -group, 292 H-group, 274 1-group, 99 :-group, 81, 215 %-group, 24 GSJP, 134 Guralnick, 88 half open intervals, 4 Hall subgroup, 252, 310 Hall :w -subgroup, 312 Hall, P., 167 Hall–Witt Identity, 96 Hall's theorem, 311 Hamiltonian group, 169 Hasse diagram, 37 higher center, 297 higher commutators, 307 higher images, 140 holomorph, 177 homomorphism, 46, 105 H-homomorphism, 274 Hopfian, 147 hyper-c , 289 hyperoctahedral group, 190

idempotent, 76, 107 identity, 20 ignore-^ map, 172 image sequence, 140 indecomposable, 75 independent, 332 index set, 152, 181 index, 61, 115 induced action, 211 induced inverse map, 105 induced map, 105 infinite cycle, 192 infinite dihedral group, 129 infinite order, 21 infinite, 12, 20 inherited, 116 injection map, 154 injective, 359 inner automorphisms, 30 insertion rules, 323 interleaved series, 294 intersection, 115, 294 invariant factor decomposition, 160, 357 invariant factors, 160 invariant factors, 357 invariant, 11, 105, 204 inverse, 20 inversion, 204 involution, 21 is direct, 75 isomorphic, 46, 106 H-isomorphic, 277 isomorphism invariant, 46, 116 isomorphism theorems, 112 H-isomorphism theorems, 276 isomorphism, 46, 106, 189 H-isomorphism, 274 isotone, 5 join, 4 Jordan–Ho¨lder Theorem, 281 ^-group, 291 ^-radical, 315 ^-residue, 315

Index ^-subgroup, 291 ^-series, 292 ^-universal property mapping, 326 ^-universal, 326 ^8 -class, 292 ^8 -group, 292 ^= -class, 292 ^= -class, 292 Kappe, 89 kernel sequence, 139 kernel, 108 Kertész, 165 Klein, 24 Krull–Remak–Schmidt Theorem, 286 Lagrange's theorem, 42 largest, 4 lattice, 8 law, 320, 327 least upper bound, 4 left action, 229 left coset, 41 left inverse, 173 left invertible, 173 left regular representation, 124, 212 left-inverse property, 359 length, 2, 77, 323 lifted, 111, 319 linear order, 3 linearly ordered set, 3 :-local subgroup, 265 locally cyclic, 59 locally finite, 59 lower bound, 4 lower center, 299 lower central series, 299 MacDonald, 89 maximal condition on subgroups, 60 maximal condition, 6, 137 maximal element, 4 maximal normal, 70 maximal, 38 maximum, 4

377

mediating morphism, 111, 155, 156, 319, 326 meet, 4 metabelian, 101, 132 minimal condition, 6, 137 minimal direct summand, 163 minimal element, 4 minimal normal, 70 minimal, 38 minimum, 4 modular lattice, 101 modular law, 101 module, 355 V-module, 355 monoid, 2 monomial group, 185 monomorphism, 106 monotone, 5 Morse, 89 move, 207 multiplicative, 43 multiplicity, 1 multiset, 1 8-ary operation, 19 8: -argument, 245 8-generated, 35 8-generator group, 35 natural projection, 108 nC-group, 165 nD-group, 165 negative, 20 nilpotency class, 301 nilpotent, 107, 216, 293 nodes, 189 nongenerator, 127 nontrivial, 21 normal closure, 71 normal complement, 151, 171 normal extension, 115 normal interior, 125 normal join, 68 normal lifting, 115, 294 normal series, 77 normal, 65, 140 normality preserving, 140

37 8

Index

normalize, 82 normalizer condition, 135 normalizer, 82 nullary operation, 19 H-composition distance, 281 H-composition factors, 278 H-composition length, 281 H-composition series, 278 H-correspondence theorem, 276 H-equivalent, 277 H-factor group, 274 H-group, 274 H-homomorphism, 274 H-isomorphic, 277 H-isomorphism theorems, 276 H-isomorphism, 274 H-quotient group, 274 H-series, 276 H-simple, 279 H-subgroup, 274 H-subnormal, 277 odd parity, 194 odd, 26, 194 open interval, 3 operator domain, 274 orbit, 80, 205, 208 orbit-stabilizer relationship, 205, 209, 210 order anti-embedding, 5 order anti-isomorphism, 5 order embedding, 5 order isomorphism, 5 order preserving, 5 order reversing, 5 order, 20, 21 outer automorphism group, 120 outer automorphism, 120 1-group, 99 :-cover, 216 :-element, 80, 215 :-group, 81, 215 :-local subgroup, 265 :-quasicyclic group, 70 :-series, 216

:-standard form, 15 :-subgroup, 81, 215 pairwise essentially disjoint, 72 partial order, 2 partially ordered set, 2 partition, 10 perfect, 84, 186 periodic, 21 permutable complement, 150 permutable, 98 permutation group, 24 permutation matrix, 184 permutation, 24, 191 permute, 39 ply transitive, 203 Poincaré's inequality, 62 Poincaré's theorem, 62 polycyclic, 293 posets, 2 positive, 54 power of the continuum, 14 power, 13 preserves orientation, 51 primary cyclic decomposition, 160, 357 primary decomposition, 243 primary, 160 :-primary, 160 principal factors, 278 principal series, 278 product, 13 projection map, 75, 154 projection, 118 projective property, 351 projective, 358 proper refinement, 276 proper subgroup, 32 proper, 77 pure, 363 quasi -group, 262 :-quasicyclic group, 70 quaternion group, 48 H-quotient group, 274 quotient group, 67 quotient, 116, 294

Index V-module, 355 radical series, 306 ^-radical, 315 range-lifted, 358 rank, 325 real, 266 reduced, 54, 323 reduction, 323 refinement, 276 reflexivity, 2, 10 regular wreath product, 182 regular, 208 relation, 338 relative holomorph, 177 relators, 338 Remak decomposition, 163 Remak, 163, 283 removal rules, 322 representation map, 123, 207 residually finite, 344 residually -group, 351 residually, 351 ^-residue, 315 restricted Burnside problem, 35 restricted symmetric group, 205 restricted wreath product, 181 reverse projection, 118 reverses orientation, 51 right action, 229 right coset, 41 right inverse, 173 right invertible, 173 right-inverse property, 358 rigid motion, 50 Robinson, 132, 134, 148 f-generated, 138 scalars, 355 Schmidt, 167 Schreier refinement theorem, 281 Schreier, 281 Schro¨der–Bernstein, 12 Schur–Zassenhaus theorem, 255 SDR, 10, 208 second isomorphism theorem, 113 second H-isomorphism theorem, 276

379

self-normalizing, 135 semidecidable, 343 semidihedral group, 178 semidirect product, 151, 171 seminormal join, 68 semisolvable, 343 sequence of normal closures, 129 :-series, 216 H-series, 276 ^-series, 292 series, 77 set product, 31 K-set, 207 sign, 194 signum, 194 H-simple, 279 simple, 84 size, 1 SJP, 132 smallest, 4 solvable by radicals, 306 solvable, 77, 293, 343 special linear group, 23 Speigel, 88 splitting field, 305 stabilizer relationship, 210 stabilizer, 205, 208 stable, 207 :-standard form, 15 step, 77 Stirling numbers of the first kind, 201 Stirling numbers of the second kind, 206 strict order, 2 string, 1 strongly disjoint, 72 strongly real, 266 subdirect product, 321 subgroup generated by, 34 ^-subgroup, 291 :-subgroup, 81, 215 H-subgroup, 274 subgroup, 32 sublattice, 9 subnormal index, 130

380

Index

subnormal join property, 132 H-subnormal, 277 subnormal, 78 substring, 2 subword, 2 sum, 13, 107 superdiagonal, 303 supersolvable, 293 supplement, 127, 145 support, 25, 152, 205 Sylow :-subgroup, 235 Sylow theorems, 237 symmetric group, 24, 191 symmetry group, 51 symmetry, 10, 50 system of distinct representatives, 10, 208

unitriangular, 229, 303 universal map, 319, 326 universal mapping property, 319 ^-universal property mapping, 326 universal, 111, 155, 156, 319 ^-universal, 326 unquotient, 116, 294 unsolvable, 343 upper bound, 4 upper central series, 297 upper factorial, 201

term, 77 third isomorphism theorem, 113, 276 Thompson Theorem, 310 Three subgroups lemma, 97 top, 4 torsion element, 353 torsion free, 21 torsion free, 353 torsion subgroup, 353 torsion, 21, 353 total order, 3 totally ordered set, 3 transitive, 203, 208, 230 #-transitive, 230 transitivity, 2, 3, 10 translation, 123 transposition, 25, 192 transversal, 10, 61, 150 triangle inequality, 130 trivial class, 291 trivial group, 21 trivial reduction, 323

Weigold, 167 well ordering, 5 well-ordering principle, 5 Wielandt, 134 Wilson's theorem, 59 word problem, 343 word, 1

unary operation, 19 uncountable, 12 undecidable, 343 underlying set, 1, 19

variety, 327 verbal subgroup, 327 vertex, 155, 156 vertices, 189 Vierergruppe, 24

Zassenhaus lemma, 119, 280 zero map, 23, 106 zero, 20 Zorn's lemma, 5