THEORY OF FINITE GROUPS A Symposium
RICHARD BRAVER Harvard University and
CHIH-HAN SAH University of Pennsylvania
o W. A. BENJAMIN, INC.
New York
1969
Amsterdam
: : :::ORY OF FINITE GROUPS , . ~·.;'osium
Copyright © 1969 by W. A. Benjamin, Inc . .-\11 rights reserved
Library of Congress Cata10g Card Number 69-20486 \tanufactured in the United States of America ;23.+5M32109 The manuscript was put into production on November 8, 1968; this volume was published on February 15, 1969
\\. A. Benjamin, Inc. "ew York, New York 10016
A Note from the Publisher This volume was printed directly from a typescript prepared by the editors, who take full responsibility for its content and appearance, The Publisher has not performed his usual functions of reviewing, editing, typesetting, and proofreading the material prior to publication. The Publisher fully endorses this informal and quick method of publishing conference proceedings, and he wishes to thank the editors for preparing the material for publication.
Preface These proceedings consist of abstracts of the lectures presented at a regional conference on finite groups held from May 2 to May 4, 1968, at Harvard University.
In addition, a number of
colleagues who were not able to lecture on their recent works at this conference were kind enough to submit abstracts to our collection.
We hope that
these papers may give a picture of some of the recent progress in this field.
We would like to acknowledge our appreciation to the financial assistance rendered by Harvard University, Massachusetts Institute of Technology and the National Science Foundation which made th conference possible.
We would also like to thank
W. A. Benjamin, Inc. for publishing these reports.
Finally, without the care and patience of Misses Taffy Jones and Mary Vallery, the manuscripts could never have been prepared for camera copy. R. B. C. H. S.
vii
TABLE OF CONTENTS
Part I Page
1.
On groups with quasi-dihedral Sylow 2-subgroups, I.
2.
21
odd.
=:
1
PSP4(q)
W. J. Wong
(mod 4)
, 31
A characterization of q
and
25
On a characterization of q
7.
H. Bender
P. Fong
D4 (q)
6.
13
On a characterization of 2
5.
R. Brauer
Finite groups having a strongly embedded subgroup.
4.
1
On groups with quasi-dihedral Sylow 2-subgroups, 11.
3.
J. L. Alperin
L (q) 4 K. W. Phan
for 39
Finite groups with abelian 2-Sylow groups.
J. H. Walter
ix
43
x
Contents
8.
A theorem on Jordan groups.
9.
On the alternating groups.
10.
N. Ito
47
T. Kondo
49
Uniprimitive permutation groups. L. L. Scott
11.
Some new simple groups of finite order.
z. 12.
Janko
63
On Janko's simple group of order 50,232,960.
13.
55
G. Higman and J. McKay
The simple group of order
604,800.
M· Hall, Jr. and D. Wales
14.
79
Le groupe de Janko d'ordre
604,800.
J. Tits
15.
91
Linear groups of degree Hall-Janko group.
16.
6
and the
J. H. Lindsey, 11
97
On the isomorphism of two groups of order
17.
65
44,352,000.
C. C. Sims
A simple group of order J. McLaughlin
101
898,128,000. 109
xi
Contents 18.
A simple group of order
448,345,497,600.
M. Suzuki
19.
113
Some simple groups related to
M . 24
Dieter Held
121
Part 11 1.
An analysis of group representations. S. B. Conlon
2.
125
Centralizer rings and characters of representations of finite groups. W. Curtis and T
3.
131
Some properties of the Green correspondence.
4.
V. Fossum
W. Feit
139
1ndecomposable modules for finite groups. G. J. Janusz
5.
Relative Grothendieck group. and I. Reiner
149
T. Y. Lam 163
xii 0.
Contents Isometries and characters of finite groups. W. F. Reynolds
7.
171
On the integral representation of an order.
H. Zassenhaus
181
Part III 1.
The order of finite Chevalley and Steinberg group.
2.
N. Iwahori
203
The Steinberg character of a finite group with a BN pair.
4.
L. Solomon
223
On the centralizer of involutions in finite groups.
6.
213
A sufficient condition for p-stability. G. Glauberman
5.
195
On some properties of groups with BN-pairs.
3.
R. Carter
D. Gorenstein
227
Centralizer of involutions in finite simple groups.
On F-projectors.
J. H. Walter
239
B. Huppert
243
Contents 8.
Locally finite versus finite simple groups.
9.
la.
xiii
O. H. Kegel
Automorphic Algebras.
247
Ernest Shult
251
Automorphisms of finite groups. C. H. Sah
261
NOTE: Pages 158-162 are not missing from this publication. Pages were numbered incorrectly when the final manuscript was produced.
ON GROUPS WITH QUASI-DIHEDRAL SYLOW 2-SUBGROUPS, I
J. L. Alperin
Our object is the ultimate classification of another class of simple groups, namely, those with a Sylow 2-subgroup
S
defined by generators
a
2
n
of order a, b
b -1
+ 1, n ~ 3,
2
and relations
+
2
n
-
1
The only simple groups known with such quasidihedral Sylow 2-subgroups are q
~
-1
(mod 4), U (q) 3 (The group
linear group
L (q) 3
with
q
L (q) 3
=1
with
(mod 4)
and
is the projective special
PSL(3, q), the group
projective special unitary group 1
U (q) 3 PSU(3, q)
is the and
FINITE GROUPS
2
M ll
is the Mathieu group on eleven letters.)
The
main result is a major step towards a classification of all simple groups with quasi-dihedral Sylow 2-subgroups.
This work has been done jointly
with R. Brauer and D. Gorenstein; the final draft is in preparation and is still subject to change.
MAIN THEOREM.
Let
G
be
~
simple group
with quasi-dihedral Sylow 2-subgroup and let be an involution in
-
either
C(x)
is isomorphic to
GL(2, q)
GU(2, q) with
(i i)
If
either
q
3
and
with
q -
~
guotient of
q = -1 (mod 4)
q - 1 (mod 4)
group of odd order
or
G .
-
(i)
x
£y
~
or
central sub-
d .
c
(mod 4),
6
+ 1
then
3
IGI =
7920 •
This result is very close to a classification. In fact,
if
q
;:
-1
(mod 4)
then this theorem and
a previous one of Brauer [1] isomorphic to d
=
(3, q
-
L (q) 3
1) .
or
imply that
is
G
and that
M
n
The case that
-
q
1
(mod 4)
is
still open but it is possible one may show that G
is a doubly transitive group of a special sort.
Hopefully, one may also prove that 3
(i t
should be
(3,
q + 1)
).
d
is
1
or
If this can be
accomplished then a result of Suzuki [3] may be applied to handle the case that
d
=
1 .
The proof of the Main Theorem is quite long and divides into three major stages.
The methods
are local group-theoretic and character-theoretic. We shall concentrate here on the former: for the latter the reader should see the paper by Brauer in this collection. these three parts.
We shall now survey each of
4 1.
FINITE GROUPS General results Various previous classification theorems
allow us to give detailed results on the structure of subgroups of groups Sylow 2-subgroups. x N
G
For example, if
is an involution of
= N/<x)
then
with quasi-dihedral
N
G
G, N = C(x)
is simple,
and
has dihedral Sylow 2-subgroups
so the results of Gorenstein-Walter [2J can be applied.
This yields that
has a series of
N
normal subgroups
N
::J
L
::J
L
o
::J
S (N)
::J
°
(N)
::J
1
with the following properties: (i)
O(N)
odd order in (ii)
N
S(N)
(iii) for some odd (iv)
is the largest normal subgroup of
=
LO/S(N)
X),
IS(N)
: O(N)
is isomorphic with
1=
2
L (q) 2
q.
2
and
L'
is odd.
•
5
(v)
q ;: -1 (mod 4) minant (mod 4) minant
is isomorphic to
L/O(N)
+1
or
+
SL- (2,
q)
if
(the group of matrices of deter+ if q ;: 1 -1) or to SU- (2, q)
(the group of unitary matrices of deter+1
or
-1).
This result is also very important as it supplies us with an invariant
q, which we call
the characteristic power, at the very beginning of our analysis.
It is, in fact, well-defined as the
assumption of simplicity forces
G
to have exactly
one conjugacy class of involutions. An important class of groups is the class of QD-groups; these are the groups H/O(H)
H
such that
has a normal subgroup of odd index which
is a simple group with quasi-dihedral Sylow 2subgroups.
This simple section is uniquely deter-
mined so its characteristic power is called the characteristic power of the prime dividing regular if
q
H.
Moreover, if
then we say that
H
p is
is
FINITE GROUPS
6
(le I ) 2 ~ P
IH I
P
N
~here
(IN
I )3 P
is the centralizer of any involution of
e
(they are all conjugate) and
~
izer of a four subgroup of with
q - -1 (mod 4)
(mod 4) G
and
are regular and
H.
The groups
U (q) 3 M ll
is the central-
with
q
=
L (q) 3 1
Indeed, if
is not.
satisfies the conclusions of the Main Theorem
and
q
1
3
then
G
is also regular.
The characters of
G
can also be studied;
using well-known methods one can examine the block structure and derive a number of formulae for the group order.
(Again, see the paper of Brauer.)
In addition, the following key result is obtained: Theorem A. of
~
QD-group
If G
H
and
is
Hand
characteristic power then O(H)
H
n
~
G
regular QD-subgroup G
have the same
is regular and
0 (G)
In particular, if proper subgroup of
G
G
is simple,
H
is a
and the conclusions of the
7
Main Theorem hold for all proper simple sections of G
with quasi-dihedral Sylow 2-subgroups then
O(H)
=
1.
Moreover, the condition on equality of
characteristic powers is equivalent with the following conditions:
H
has exactly one conjugacy
class of involutions and if H
then
H
is an involution of
is of odd index in
CH(x) O(C(x»
This condition on
x
C(x)
is less stringent then that
for strong embedding, namely
CH(x)
=
C(x), or
even for that of weak embedding, namely, C (x)
=
C (x) 0 (C (x) ) H
As in other classification results, one tries to prove that the subgroup structure of certain form.
G
has a
On the assumption that it does not,
much of the argument is devoted to producing such a subgroup
H
in which
dieting Theorem A.
O(H)
f
1,
and so contra-
Because of that result we do
not need to construct a strongly embedded subgroup, as was done in previous problems, and consequently we do not need to establish any so-called Etheorems, which saves considerable effort.
FINITE GROUPS
8
2.
Structure of
O(N)
In proving the Main Theorem by induction the bulk of the local group-theoretic analysis is devoted to establishing the following result: Theorem B.
Let
G
be.§. simple group with
guasi-dihedral Sylow 2-subgroup and of characteristic power of
G
If every proper simple section
q
with quasi-dihedral Sylow 2-subgroup sat is-
fies - -the -
conclusions of-the - -Main - - Theorem and
is the centralizer of an involution of (i)
O(N) = AB
of order dividing 6 B
±l),
f
nel
1
then
+ 6 2
A
N
for
then
is.§. cyclic subgroup and
(q '" 6 (mod 4)
is an abelian subgroup and if. A O(N)
Any
1,
A .
~-central
involution of
non-identity element
N
inverts
A .
The extended centralizer ~
f
is .§. Frobenius group with ker-
and centralizes some conjugate of (iii)
in
q
Band complement (ii)
B
B
where
G
N
C*(u) u
of
lies O(N)
This result also has a consequence the fact
9
that the subgroup
La
direct product of
O(N)
with
SL(2, q)
if
q
of
~
N
defined above is the
and a subgroup isomorphic 9.
When
one other possible structure for
q La
=
9, there is
as
SL(2, 9)
has a non-trivial central extension by a group of order
3 The first step is to reduce the proof of
Theorem B to the proof of the following result (with same hypotheses) : Theorem B*.
10 (N)
1
and
If
P
is
~
prime divisor of
R
is a Sylow p-subgroup of
is
~
0 (N)
then (i) (ii)
R R
Sylow p-subgroup of
O(N(R»,
is centralized or inverted £Y some
non-central involution of
N
This reduction can be established quickly even though Theorem B and Theorem B* appear to be so dissimilar. The focus is thus shifted to the plocal subgroups of
10 (N) I.
G
for primes
p
dividing
We first note that the case of character-
FINITE GROUPS
10 istic power
3
has been dealt with by Wong [4J,
who showed that
and
L (3) 3
such simple groups.
M ll
are the only
Thus, We may assume
q
f
3
The proof consists of a long demonstration that if p
is a prime for which Theorem B* fails then there
is a p-local subgroup characteristic power theses
H
H
which is a QD-subgroup of
q.
is regular as
However, by our hypoq
f
3
so
H
violates
Theorem A. The construction of such subgroups
H
re-
quires the usual "pushing-up" methods for constructing p-local subgroups, namely, transitivity theorems for signalizers and Glauberman's theorem.
Z(J)
Actually, we require extensions of both
these ideas.
First, Glauberman's result can be
extended to certain p-subgroups other than Sylow p-subgroups.
Second, we define the notation
1ft H(X, Y; for subgroups
H, X, Y
p)
and a prime
collection of all elements of
p to be the
"" H(X, p)
(that is,
11
p-subgroups of
H,
normalized by
X
and inter-
secting
X
with
and we prove transitivity theorems for
Y
trivially) which are also permutable
the maximal elements of 2-groups
3.
X
~H(X,y;P)
and particular subgroups
The structure of
for certain Y.
N
The final part contains the transition from the results of Theorem B to the Main Theorem.
The
proof is a combination of group-theoretic and character-theoretic techniques, but mainly the latter ones.
The information provided by Theorem
B for an inductive proof of the Main Theorem can be used to derive sharpened group order formulae (see Brauer's talk). that
N
=
Land
The main steps needed are proofs B
=
1 .
FINITE GROUPS
12 REFERENCES
1.
Brauer, R., On finite Desarguesian planes I, II, Math. Zeit., 90 (1965), 117-123; 91 (1966), 124-151.
2.
Gorenstein, D., and Walter, J. H., The characterization of finite groups with dihedral Sylow 2-subgroups, I, II, III, ~. of Algebra, 2 (1965), 85-151, 218-270, 334-393.
3.
Suzuki, M., A- characterization of the 3dimensional projective unitary group over a finite field of odd characteristic, J. of Algebra, 2 (1965), 1-14.
4.
Wong, W., On finite groups whose Sylow 2have cyclic subgroups of index 2, ~. Australian Math. Soc., 4 (1964), 90-112.
ON GROUPS WITH QUASI-DIHEDRAL SYLOW 2-SUBGROUPS, II
Richard Brauer
This paper is a continuation of the preceeding paper to which we refer as I.
We shall report in
more detail on the character-theoretic part of our joint work.
Throughout, the same notation is used
as in I. 1.
Consider a finite group
dihedral Sylow 2-subgroup and assume that group of index
G 2.
S
G
with quasi-
of order
n 2 + 1
does not have a normal subIt has already been shown in
Brauer [5] Section VIII that the principal 2block
BO(G)
of
G
consists of
irreducible characters. Xl '
X2 '
X 3
n 4 + 2 - 1
Four of them, say
X
o
have odd degrees and the notation 13
l~
FINITE GROUPS
can be chosen such that characters in
BO(G)
X
o
=
that the remaining character of
signs.
1
n -
Xl (1) + €l
have degree
BO(G)
and
Here,
2
1 , that
- 1
and
has degree
are suitable
Moreover, we have a formula of the follow-
ing form
IN
3 1
(1 )
lel Here, tion
in the center of
tralizer Moreover, by
(Xl(l)
-
Nand
CG(T) q
6€lq)
e
is the centralizer
N
x
2
Xl (1) (Xl (1) + €l)
G
(x)
q
of the involu-
e
Sand
+ 6
q
2
of a four-subgroup
is the cenT
of
S .
is an odd integer uniquely determined 6
=
±l
and
q
~
6
notation of Brauer [4], we have
(mod 4) q
=
(In the
-m6.)
A
great deal further information is available concerning the characters of
G .
On comparing these results with those in I, Section 1, it appears that "characteristic power".
q
actually is the
In particular,
q
is a
15
power of an odd prime p . If
G
is core-free, i.e. if
then for given given.
q
The case
O(G)
IG I
upper bounds for
=
q
3
=
1 , can be
can be disposed of, as
this has been done in Wong [4] and we may assume q
>
3 •
2.
We shall say that our group
lar, if the prime factor
p
of
q
G
is regu-
with at least the same exponent with which divides
\N\3/
Ic 12
If
.
G
IGI
divides p
is regular, it can
be shown that
(2)
X (1) 3
=
Xl (1)
q(q
2
-
q
3
X (1) 2
=
q
2
- 5q + 1 ,
5q + 1)
Conversely, each of these three equations implies regularity.
For regular
(q
2
G
- q5 + 1)
(3) (q -
0)
2
(1) reads 2
16
FINITE GROUPS 3.
We are now ready to prove Theorem A of I.
Suppose that
G
is a QD-group which contains as
a subgroup a regular QD-group characteristic power.
If
same significance for
G
H
of the same have the
as above, it is possible
to show that at least one of the characters remains irreducible when restricted to that
Xi[H
plays in
plays the same role in Ba (G)
shows that
G
X.
1-
Hand
BO(H)
as
X.
1-
Now the remark following (2) is regular.
Combining the results
obtained with general properties of blocks, we obtain the last part of Theorem A. 4.
In principle, each 2-block of
used to obtain formulas for
IG
I.
G
can be
The necessary
techniques have been developed in Brauer [5J, [6J. In particular, we use blocks with the defect group T
and special blocks with the defect group
S
In this manner we obtain sufficient conditions for a QD-group to be regular. tion concerning the number
We also obtain informaIN: L
I •
We don't know how far one can get by applying
17 character theoretic methods and using the information on
N
given in I, Section 1, first paragraph.
It seems that the discussion would become very complicated.
It is now extremely fortunate that
after Theorem A has been obtained, it is possible to prove Theorem B of I by a local group theoretical analysis.
No character theoretic methods are
needed for this step. In attempting to prove the Main Theorem of I by induction, we may then assume that the conditions stated in Theorem B. further information on N
N
satisfies
Also some
becomes available.
This
results in a decisive simplification in applying the character theoretic methods mentioned at the beginning of the present section.-
= L .
possible to show that
N
criteria show that
is regular.
regularity of
G
G
implies that
It now becomes
If
B
A
f
1 , our
In addition,
=
1
It remains to deal with the case
A
=
1 .
Here the character theoretic arguments show that divides
q
2
- 1 .
Now the information
FINITE GROUPS
18
available from the local group theoretic analysis implies that
B
=
1.
Finally character theoretic
arguments of the type used in Brauer [lJ, II, Section 10 shows that if q ;
7.
Moreover,
G
A
=
B
=
1
we must have
is regular for
q
>
3
Now, the proof of the Main Theorem of I can be completed.
In particular, the formula for
IGI
is
obtained from (3). 5.
As was mentioned in I, the Main Theorem
of I shows that, for a simple QD-group characteristic power (i.e. with
6
=
are satisfied. with
L (q) 3
or
with
q
~
-1
of
(mod 4)
-1) the assumptions of Brauer [lJ It follows that M ll
is not regular and In the case
q
G
G
is isomorphic
In the latter case, q
=
G
3 .
q - 1 (mod 4)
questions are still open.
, the corresponding
Our results allow us to
determine a large part of the character table of G.
It seems likely that a continuation of this
work will yield the complete character table of G.
Because of lack of time, this step has not
19 been carried out.
In any case, a great deal of
information on
will be available.
G
hope that one can prove that
G
There
lS
is isomorphic to
REFERENCES
[lJ
-
[4J
are the papers quoted in I.
[5]
Brauer, R., Some applications of the theory of blocks of characters of finite groups 111, J. of Algebra 3 (1966), 225-255.
[6J
Brauer, R., On blocks and sections in finite groups, American J. Of Math. 89 (1967), 1115-1136.
FINITE GROUPS HAVING A STRONGLY EMBEDDED SUBGROUP
Helmut Bender
Following [1], I call a proper subgroup of a finite group if
H
G
strongly embedded (in
has even order but, for every
g H n H
H G)
g EG - H ,
has odd order.
Theorem.
Let
G
be a finite group having a
strongly embedded subgroup
H
Then (i) or (ii) holds: (i)
G
has a normal series
such that G /G 2 l
G l
and
G/G
2
have odd order and
is isomorphic to one of the simple groups:
PSL(2,q), Sz(q), PSU(3,q), 21
where
q
=
n
2
>
2 .
FINITE GROUPS
22
(ii)
A Sy10w 2-subgroup of
G
is cyclic or
a (generalized or ordinary) quaternion group. It suffices to assume that
H
is a proper sub-
group of even order which does not contain all involutions of involution J
=
(x
G
and that
x EH
then
E HI x
group of
2
= 1J
CG(x)
EH
for every
NG(J), with
is a strongly embedded sub-
G .
The theorem generalizes Suzuki's c1assification of (TI)-groups, see [3J. Several important general properties of
G
are well known from some of Suzuki's papers, for example: For every in
H
g E G,
g H n H
permutes involutions
transitively.
These properties together with the following proposition form the starting point for the induction proof of the theorem: Let
p
be a prime and
p-subgroups of
H
n
= 2
or
3 .
Among
which are contained in at
23
least
n
conjugates of
H, let
P
be maxi-
mal. If
n = 3, then
If
n = 2, and
NH(P)
is not doubly transitive
G
on conjugates of
has even order.
H
,
then again
has
NH(P)
even order. If
X
tained in then X.
X
is a proper subgroup of such that
H
n
H
X
nH
G
not con-
has even order,
is a strongly embedded subgroup of
Proceeding by induction, one obtains some
information about
X .
It is the aim to show that transitive on conjugates of g E G,
H
n Hg
H
G
is doubly
and, for every
has a normal complement in
H
Then a theorem of Suzuki [2J comes into action and yields the conclusion of the theorem above. The just mentioned proposition allows to prove doubly transitivity at a rather early stage, and it provides one with sufficiently many subgroups to which induction hypothesis can be applied.
FINITE GROUPS
24
REFERENCES
:1]
D. Gorenstein, Finite groups; Harper and New York 1968.
-2
M. Suzuki, On a class of doubly transitive groups: 11; Ann. of Math. 79 (1964), 514-589.
-
"'
M. Suzuki, Finite groups of even order in which Sylow 2-groups are independent; Ann. of Math. 80 (1964), 58-77.
Ro~,
ON A CHARACTERIZATION OF 2
AND
D (q) 4
Paul Fong
Let q
nl
== Pl 1
be odd primes, and let
Pi' P2
,
spectively. SL (2, q. ) l
,
product of
n2 P2
=
q2
For
i
be powers of
=
,
1, 2
let
and let
Co == C1 C 2
Cl
C
and
an extension of order
C. l
be a copy of
be the central
Finally, let
2 2
over
CO.
to the group
j C.
be denoted by satisfying
such that
(*)
CG(j)
This property
(*)
C
be
We are in-
terested in investigating simple groups an involution
re-
Pl' P2
G
with
is isomorphic CG(j) 2 C
will
The known simple groups
are the four-dimensional projective
symplectic group, the Dickson group [2J, and the 25
FINITE GROUPS
26
"='.visted Steinberg group [3 J defined over a finite :ield of odd characteristic. :.:~ese
In the Lie notation,
are the groups
1 q 4 (4 or d er 2 q - 1) (2 q - 1) ,
) B ( q, 2
662 order q (q - 1) (q - 1), 2
D (q), 4
order q
12
6
2
q 8
4
(q - 1) (q - 1) (q + q + 1),
Wong [4J has shown that if
G
is a simple
(*), then necessarily
group satisfying Moreover,
q
C(j)
Pl
=
P2
G::::. B (q), or C1
Our result
is concerned with identifying the groups
G
in
this latter case. Theorem:
Let
G
be a group satisfying
Suppose Cl <::I C, C
G
f
C(j)O(G)
p.
has one of two distinct structures: in which case
and
=
If
q = min (ql' q2}
>
9, then
27
either
q
(ql' q2}
= q1 = q 2 3 = (q , q}
and and
G:::G
2
(q),
or
2 G ~ D 4 (q) .
As in other characterizations, the proof develops along what has now become established lines (see for example [1], [4J).
We mention two
points: 1)
In constructing a subgroup of
G
with a
suitable (B,N)- structure (the subgroup being in fact the group We want
G
to be isomorphic to), divides
it is essential to know that
IGI .
This can be shown by a consideration of the
2-blocks of
G
with a four-group as defect group.
If such a block has one irreducible modular character, then the character formulas available in this situation give the desired divisibility.
If every
such block has more than one irreducible modular character, then
q
~
In this last part we
9 .
require the analysis carried out by Brauer in [lJ, §loo 2)
That
and
q2
can only be
q
or
q
3
28
FINITE GROUPS
is shown as follows:
G
has a section V
which
is an elementary abelian p-group on which
C., i
E (1,
Cv (j)
1
~
of
SL(2,
2},
acts.
The action is such that
Now the irreducible representations q.) ~
over the prime field
F
p
have been
listed, and an inspection of this list shows that the dimension of 8
- n 3 . ~
,
V
over
or not less than
F
p
4n.
~
must be
2n.
~
,
The first case is
actually impossible, the second case occurs only if 3
divides
n.
~
and leads to
2
case occurs only if the dimension of and leads to
G (q) 2
the third
D4 (q)
V
is 4n.
~
29 REFERENCES
1.
R. Brauer, On finite Desarguesian planes, 11, Math. Zeit., Vol 91 (1966), 124-151.
2.
L. E. Dickson, Linear groups in an arbitrary field, Trans. Amer. Math. Soc. Vol 2 (1901), 363-394.
3.
R. Steinberg, Variations on a theme of Chevalley, Pac. J. Math., Vol 9 (1959), 875-891.
4.
W.J. Wong, These proceedings.
A CHARACTERIZATION OF
W. J. Wong
1
We consider finite groups
G
with the
following property.
(*)
Lln L
If
q
G
2
has subgroups
= <
t
>
L , L 2 l
,where
t
is odd, the simple group
property, with
ql = q2 = q .
has order 2,
PSP4(q)
q
has this
In this case we
write a 4-dimensional vector space field of
such that
V
over the
elements, with a non-degenerate
1. Research partially supported by National Science Foundation grant GP-6652 at the University of Notre Dame.
31
32
FINITE GROUPS
alternating bilinear form,
as an othogonal direct
sum of two hyperbolic planes L , L 2 l
as the subgroups of t~e
ing to
correspond-
PSP4(q)
subgroups of the symplectic group leaving fixed the vectors of
SP4 (q) = Sp (V)
V ' V 2 l
and take
V ' V ' 2 l
respectively.
The extent to which ized by the property
PSP4(q)
(*)
is character-
is indicated by the
following result. THEOREM.
Let
property (*)
G
be a finite group with the G ~ C(t)O(G)
such that where
u
2
= 1
Then
and one of
the following holds. ( i) L u 1
=
L
has one class of involutions,
G
,
1
L
u 2
=
L
2
,
u
phisms of linear type on and
IL
2
\
and
induces outer automorL
l
and
L
2
,
ILl \
are divisible by the same powers of and
are powers of the same prime
:-:c:mber. (ii)
G
has two classes of involutions,
2,
33
and
Here of G
G
O(G)
having odd order; the condition that
+ C(t)O(G)
simple.
is the largest normal subgroup
is obviously satisfied if
By an automorphism of
G
is
of linear
SL 2 (q)
type we mean one induced by an element of GL 2 (q) . Since all involutions of PSL (q) 2
PGL (q) 2
not in
are conjugate, the structure of
in case (i)
is uniquely determined.
occurs in the simple groups
This case 2
G2 (q), D4 (q)
and is discussed in Paul Fong's article. mark incidentally that the condition that
and
are odd, since
C(t)
(q odd), We reimplies
and
have non-trivial centers. We shall sketch the proof of the theorem.
A
theorem of Glauberman [4J shows that the condition G
+C(t)O(G)
lS
is conjugate in
equivalent with saying that G
t
to some other involution of
FINITE GROUPS
34
A study of the Sylow 2-subgroups of the
C(t)
centralizers in that
C(t)
C(t)
of various involutions shows
has one of the two possible structures
indicated in the theorem, and that the number of classes of involutions in
G
is correspondingly
one or two. In case (i), suppose that
ql
and
powers of different prime numbers. ql
>
q2
number
and that p.
q2
q2
are
Suppose that
is a power of the prime
We apply methods and results of
Brauer [2,3J
(on groups whose Sylow 2-subgroups
are quasi-dihedral or the wreath product of a cyclic group and a group of order 2) to centralizers in
G
of certain elements of odd order, in
order to show that a Sylow p-subgroup of is not one of
G.
a certain section
C(t)
This implies the existence of V
in
G
Abelian p-group and on which
which is an elementary
L
l
acts non-trivially.
The degrees of the irreducible Brauer characters of
for the prime
p
being known, we find a
lower bound for the dimension of
V
as a vector
35
space over the field of
p
elements.
On the
other hand, there is a four-group acting on
V
and we can find an upper bound for the dimension. The two bounds are compatible only when q2 = 3.
Then
V
L
l
5 ,
is the Frattini factor group of 3
an extra-special group of order action of
=
ql
on
V
5
,
and the
corresponds to an embedding in such a way that an
element of order
3
leaves the vectors of a
hyperbolic plane fixed. such embedding of
However, there is no
SL (5) 2
This
contradiction shows that
and
must be
powers of the same prime number. The identification of
G
with
PSP4(Ql)
in
case (ii) was obtained by Janko in the case Ql
=
3
[6J.
general case.
Similar methods can be used in the First we determine the structure of
the centralizer in gate to
t
G
of an involution not conju-
This is done by transfer methods, a
theorem of Gorenstein and Walter on groups with dihedral Sylow 2-subgroups [5J, and a
formula
FINITE GROUPS
36
of Brauer on the numbers of fixed points of elements of a four-group acting on a group of odd order [1].
We find that the centralizer is a
split extension of a normal subgroup isomorphic with
PSL (ql) 2
2(ql - €)
by
where
a dihedral group of order ql
=0
(mod 4),
€
Having done this, we set up a for
G
?-subgroup of ?rime number
(BN)-pair
in
G
G, where
ql
is a power of the
p.
of a Sylow
This is done by determining
centralizers and normalizers of various smaller p-subgroups of
C(t)
main tool is the Brauer formula referred to
320ve. a~d
±l .
B
?-subgroups, beginning with ~~e
=
Most of the work is involved in deter-
Dining the normalizer
~~e
€
Because of the abundance of four-groups
particularly the existence of two classes of
~~volutions,
we do not have to use directly any
arguments from the theory of group characters. ~~entually ~or~alizer
s~o·.,:
we determine the exact structure of the B
of a Sylow p-subgroup of
G
that together with a certain subgroup
and N
of
37 C (t)
it forms a
then
GO
(BN)-pair [7].
is a subgroup of
order of
G
If
=
GO
BNB ,
whose order is the
and the structure of
uniquely determined. Since
is
It follows that GO
has two classes of
involutions, and contains the centralizer in of each of its involutions, we have
GO
=
G .
G
FINITE GROUPS
38
REFERENCES
1.
R. Brauer, Some applications of the theory of blocks of characters of finite groups 11, J. Algebra 1 (1964), 307-334.
2.
R. Brauer, On finite Desarguesian planes 11, Math. Z. 91 (1966), 124-151.
3.
R. Brauer, Investigations on groups of even order 11, Proc. Nat. Acad. Sci. USA 55 (1966), 254-259.
4.
G. Glauberman, Central elements in core-free groups, J. Algebra 4 (1966), 403-420.
5.
D. Gorenstein and J. H. Walter, On finite groups with dihedral Sylow 2-subgroups, Illinois J. Math. 6 (1962), 553-593.
6.
Z. Janko, A characterization of the finite simple group PSP4(3), Canad. J. Math. 19 (1967), 872-894.
7.
J. Tits, Theoreme de Bruhat et sous-groupes paraboliques, C. R. Acad. Sci. Paris 254 (1962), 2910-2912.
A CHARACTERIZATION OF FOR
K.
q
w. Phan,
'" -1
(mod 4)
Bonn (Germany)
The following result has been proved: THEOREM:
Let
t
o
be an involution contained L (q) 4
in the center of a Sylow 2-subgroup of where
q
H
o
Let
is congruent to the centralizer of G
be
~
-1 t
4.
modulo o
Denote
in
finite group with the following
properties: (a)
G
has no subgroup of index 2,
G
has an involution
and, (b)
that the centralizer of
t
in
G
t
C (t) = H G
is isomorphic to 39
such
H
o
FINITE GROUPS
40 Then
G
is isomorphic to
L (q) 4
The methods of the proof are group-theoretic. Our main aim is to construct two subgroups of fulfilling the properties of a sense of Tits.
G
(B,N) pair in the
Later we show that these two sub-
groups generate a subgroup
G o
of
and finally prove that
isomorphic
G
G
o
=
G .
To
this end we begin with the study of the structure of
H, in particular the determination of the
number of classes of involutions in centralizers.
H
and their
Straight-forward arguments and a
result of Thompson show among other things that G
has two classes of involutions and is a simple
group. H
The manner in which the involutions of
fuse together in
G, the first theorem of
Grun and a result of Gorenstein-Walter are used to determine the structure of the centralizer of an involution not conjugate to
t
in
G .
From the structure of the centralizer of an involution of the second type we obtain a series of results on
G
It is then possible to
41 determine the structure of a Sy10w p-subgroup of G
and its normalizer in
divisor of
q.
where
p
is the
The Sylow p-normalizer
the normalizer
N
of a complement in
Sy10w p-subgroup of G
G
B
mentioned earlier.
Band B
to the
are the two subgroups of With the structure of these
two subgroups and the action of some elements of N
on some subgroups of
to show that
BNB
~
B
L (q) 4
known, where
BNB
union of the set of double cosets in
N
that
it is then easy
BnB
is the with
n
Applying a result of Thompson shows
=
BNB
G
completing the proof.
We hope to show that the theorem is true for q
=
4h + 1
where
h
is an integer.
Some of the
main results obtained so far are: 1.
The centralizer
H
of
t
in
G
is a
non-splitting extension of the central product of two subgroups isomorphic to hedral group of order of
q
= 2.
-1
(mod 4)
When
q
=5
SL(2, q)
2(q - 1).
by a di-
(In the case
the extension splits). (mod 8)
G
has one class of
42
FINITE GROUPS
involutions.
that
3.
The group
4.
A Sylow p-subgroup of
5.
There exists a subgroup
NG(K)/K
G
is simple. G
has order K
of
G
q
6
such
is isomorphic to the symmetric
group on 4 letters. A similar characterization of
U (3) 4
has
been done and will appear in the Journal of the Australian Math. Society.
It is hoped that the
methods can be generalized to the case of wi th odd
q
U (q) 4
FINITE GROUPS WITH ABELIAN SYLOW 2-SUBGROUPS
J. H. Walter
The purpose of this note is to announce a result obtained last year but not yet published. Let
G
be a finite group.
Denote by
O(G)
maximal subgroup of odd order.
Denote by
the minimal normal subgroup
of
G/N
N
G
the 0' (G)
for which
has odd order. THEOREM.
abelian
Let
G
S2-subgroups.
direct product of
~
be
~
finite group with
Then
0' (G/O(G))
is the
2-group and simple groups of
the following types: n 2 ) ,
(1)
PSL(2,
(2)
PSL(2, q)
q
>
3
n
>
, q - 3
,
43
1 or
5 (mod 8)
,
44
FINITE GROUPS (3 )
~
simple group
involution
J
where
K
or
(mod 8)
5
of
M
such that - -for - -each --
M
is isomorphic to
PSL(2, q) , q - 3
Simple groups of the third type have not yet been completely determined.
However,
it is known
from the results of Janko and Thompson [3] that q =
5
or
q = 3
2n + 1
, n
>
In the former
0 .
case, the group is the simple group of order 175,560 discovered by Janko [2].
In the latter case the
groups are presumably the Ree groups associated with the Lie algebra istic 3.
G
2
over a field of character-
John Thompson [5] has made considerable
progress toward showing that this is the case. Their character tables have been worked out by H.N. Ward [6]. where
q
__ 32n
Their order is
q
3
(q
3
+
1) (q -
1)
+ 1
The groups of the first type are characterized in terms of conditions given by Suzuki [4] The groups of the second type are groups with a four group as an s2-subgroup and hence are
,
45 characterized by Gorenstein and Walter [lJ.
REFERENCES
1.
Gorenstein, D. and Walter, J. H. The characterization of finite groups with dihedral Sylow 2-subgroups. J. Algebra 2 (1965), 85-151, 218-270, 354-393.
2.
Janko, Z. A new finite simple group with abelian 2-Sylow subgroups and its characterization. J. Algebra 3 (1966), 147-186.
3.
Janko, Z. and J. G. Thompson. On a class of J. Algebra 4 finite simple groups of Ree. (1966), 274-292.
4.
Suzuki, M. Finite groups of even order in which the Sylow 2-subgroups are independent Ann. Math. 80 (1964), 58-77.
5.
Thompson, J. G. Towards a characterization of E *(q), J. Algebra 7 (1967),406-414. 2
6.
Ward, H. N. On Ree's series of simple groups. Trans. Am. Math. Soc. 121 (1966), 62-89.
A THEOREM ON JORDAN GROUPS
Noboru Ita
Let
G
be the set of symbols
> 3)
1,2, ... ,n
called points.
Let
G
doubly transitive permutation group on that the stabilizer and
2
3, ... , k
in
G
G l
~
,2
such
of the points 1
stabilizes further points
~3;
and moreover \I -
H
G
be a
H
lS
H
~
G
1, 2-
G
1, 2, ... , k
regular and transitive on
(1,2, ... ,k} (1,2, ... ,k}
tions in
G
and its transforms by permuta-
are called lines.
Let
A
be the
family of all lines containing the point
1.
Then the stabilizer
in
G
of the point
1
1
can be represented as a permutation group 47
G
48
FINITE GROUPS
\(G ) l
on
1\.
Now groups
G
having these properties can be
classified as follows: (1)
G
If
(n, k)
=
is isomorphic to
then
=0
n ( 2)
k
LF(3,2) 7
=
If
=
3 , n (n,k)
If
n - 0 (mod k) k
isomorphic to If
A
n
3,
+1
,
of a prime and
and if
=
n
15
A(G ) l
and
G
is
is
7
=0
not faithful, then q
>
is
(mod k) k
=
2
,
,
q n
where
=
q
2
q and
is a power G
is
isomorphic to the split extension of the 2-dimensional vector space field
GF (q)
and
7
(mod k)
faithful, then
(3 )
then
1 ,
of
q
V (2, q)
elements by
over the
GL (2, q)
ON THE ALTERNATING GROUPS
Takeshi Kondo
This is a summary of the forthcoming papers [5] and [6], which will be published in Jour. of Soc. of Japan.
M~th.
I.
Lpt
letters
(1,
where
n
Define
A
m
2,
be the alternating group on
... ,
m} •
Put
m
=
4n + r
is a positive integer and n
involutions
a
k
(1
~
k
~
0
n)
, r
~
m
of
3
~
A
m
as follows: a
Then
=
k
a
n
(1,
(4k-3, 4k-2) (4k-l,
4)
4k).
is contained in the cent er of a 2-Sylow
subgroup of note by
2) (3,
A
m
H(n, r)
For
r
=
1 ,
2 and 3 , we de-
the centralizer in
Then we have the following theorems.
49
A m
of
a
n
FINITE GROUPS
50
Theorem 1.
Let
G(n, r)
be a finite group
with the following properties: (1)
G(n, r)
has no subgroup of index 2,
G (n,
contains an involution
and ( 2)
r)
in the center of G (n,
r)
~
Then if to r
=
A
4n + r
2
where
r
=
2
or
2)
For the case
r
C
G(n, r)
(a )
n
H(n, r)
3, G (n, r)
except for the G(l,
n
2-Sylow subgroup of
whose centralizer
is isomorphic to
a
~ A
=
or
6
n
~
is isomorphic 1
and
PSL(2, 7)
1 , we have not obtained
the analogous result, but we can prove much weaker result.
We note that
H(n, 1)
elementary abelian subgroup
has a unique
S
of order
22n
up
to conjugacy. Theorem 2.
Let
G(n, 1)
be a finite group
contcining an involution whose centralizer ~
isomorphic to
H(n, 1)
abelian subgroup of
, and
H(n, 1)
S
H(n, 1)
be an elementary
of order 22n .
51 ~ ~-to-one
Assume that there exists from in
e
mapping
8
U N (S) (the set-theoretic union A m such that onto H(n, 1) U N G(n, 1) (S)
H(n, 1)
A ) m
induces an isomorphism
(resp.
(S»)
N
A
Then
and
G(n, 1)
1)
(resp. NG(n, l)(S))
is isomorphic to
(H. Yamaki)
G(3, r)
Th. 1.
H (n,
H(n, 1)
m
Theorem 3. group
between
Let
or
G
be
r
~
finite
satisfying the assumption of
Then (i)
if
r = 1 or
G
is isomorphic
l
G
has precisely four
l
conjugacy classes of involutions, and (ii) if r
=
2
or
n.
3,
G
r
Remarks.
is isomorphic to (1 )
If
r
=
Yamaki showed that
G l
A 12
r
Th. 3 1
2.
In fact,
has two possibilities
for the fusion of involutions of which is that of
+ r
12
2 or 3
is a special case of Th. 1, but if Th. 3 is a better result than Th.
A
or
A
that of the symplectic group
l3
G l
,
one of
and the other SP6(2)
1.5
(Note that
satisfies the condition of Th. 3 for
52
r
FINITE GROUPS
=
if
1.)
Then it is not difficult to see that, has the first case for the fusion of in-
G 1
volutions, (2)
G
l
satisfies the assumption of Th.
2.
The work of Yamaki was done before
Th. 1 and Th.
2 were obtained.
In fact,
a part of
the proof of Th. 1 and 2 owes to his idea. (3)
In [4; Th. A], we determined the fusion
of involutions of
G(n, r)
for the case
r
=
2 or 3.
This is equivalent to the determination of the structure of
NG(n, r) (S)
,where
mentary abelian subgroup of (If
r = 2 or 3 ,
to conjugacy.)
S
H(n, r)
is some eleof order 2
2n
such subgroup is not unique up
But we have not obtained a result
similar to [4; Th. A]
for the case
r = 1.
This
is the reason why the stronger condition is necessary for Th. t~at,
of
2.
However, we note that Th.
2 shows
if We can determine the fusion of involutions
G(n, 1)
under the same assumption as Th. 1,
we shall be ?ble to obt?in a similar result to T~
1 also for the case (4)
r = 1 .
For small value of
m,
Th. 1 and
53
Th. 2 were treated by A. Fowler, M. Suzuki and D. Held.
We used these results for the proof of
our theorem. In [3], we proved the following theorem
Ill.
which is a generalization of W. J. Wong's theorem
Theorem.
Let
G
be a finite group satisfy-
ing the following conditionr there exist volutions
a 1'
one mapping
a
2
,
... , n
from
CO
a
U
k=l theoretic union in that and
A )
m
of
n CA
G
(~k)
m onto
and a ---
isomorphic to
A
~ n)
Then if
in-
one-to-
(the setn U CG (a k ) k=l
induces an isomorphism between CG (a ) (1 < k k
n
m
>8 =
C
A m
such (a~)
k
G
is
with
A m
,
m
For the identification of
G(n, r)
we used this theorem, the proof of which is due to the idea of D. Held [1] and [2].
54
FINITE GROUPS REFERENCES
1.
D. Held, A characterization of the alternating groups of degree eight and nine, J. of Alg., 7 (1967), 218-237.
2.
-----, A characterization of some multiply transitive permutation groups I, Illinois J. (to appear).
3.
T. Kondo, On the alternating groups, J. Fac. Sci. Univ. Tokyo (to appear).
4.
-----, On finite groups with a 2-Sylow subgroup isomorphic to that of the symmetric group of degree 4n , J. Math. Soc. Japan (to appear).
5.
-----, On the alternating groups II, appear) .
6.
H. Yamaki, A characterization of the alternating groups of degrees 12, 13, 14, 15 (to appear) .
(to
UNIPRIMITIVE PERMUTATION GROUPS *
Leonard L. Scott, Jr.
A well-known result of Wielandt (lOJ is Theorem group on 2p
~
2
m
(Wielandt) 2p
+ 1
Let
letters,
be a uniprimitive
@
p
a prime.
Then
for some odd positive integer
m.
Despite the specific nature and simplicity of the condition
2p
~
2
m
+ 1 , the only known
examples of uniprimitive groups of degree p 6
a prime, occur when
5
p
~
5
acting on the 2-subsets of (1,
m
~
3
2p, ~5
and
2, 3, 4, 5) ).
The term " un iprimitive" is due to W.A.Manning * and refers to primitive permutation groups which are not doubly transitive. The results announced in this article are part of the author's doctoral dissertation submitted to the faculty of Yale University. 55
56
FINITE GROUPS
For a given prime
p, the existence of a group
which is uniprimitive of degree 2p
implies the
existence of a simple group with the same property. Wielandt showed in unpublished work that no uniprimitive group of degree despite
= 52 + 1 .
26
=
2.13
26
exist,
The author has proved that,
in fact, If
Theorem 1 2p
=
p
2 m + 1
5
m
, =
@
is a uniprimitive group of degree
where
p
and
m
3 (and
@
= m5
or
are primes, then 15
5
Feit has recently shown (unpublished) that the case
p = 41
eliminated
does not occur, and the author has p = 113.
tive groups of degree unless
p
=
Thus there are no uniprimi2p
p
=
113
p
<
313
5 .
The proof of Theorem 1 of
for any prime
and the elimination
are obtained by combining modular-
theoretic methods (see especially [lJ and [4J) with some new, more permutation-theoretic approaches. The following two definitions are especially relevant, and should prove useful in other studies
57
involving permutation groups - especially uniprimitive groups. Definition 1
~l'
Let
... ,
~r
be all the orbitals
of a transitive permutation group
@ on
~.
(The concept of an orbital is due to D. G. Higman [5].
orbit 6 (a) G
6: 0. .... 2
An orbital 6(0,)
=
of rank
~
of
r
Define the
e.l
a
has
(G)
i
r
th
Definition 2
E
Let
mutation group relative to
G E
E "
an
A transitive group
@
orbitals; see also Sims [8]).
e.l
orbital character
I(a
a
in such a way that
@
for all
(aG)
assigns to
"
n I
aG E ~. (a)}
11:= 91
~
l
I
by
(G E @)
be subgroups of the per-
Define the support of
~
U
and define a corresponding numerical relative support function s(91:U)
1~(91:U)
I .
58
FINITE GROUPS The concept of an orbital character directly
generalizes the notion of the permutation character _ ~
if we set
-1 =
e.
lll(cx) = [et}, et EO, we get
The orbital characters have several nice
o:roperties:
Each
~on-negative
integer values, and
i
> 1.
is a class function taking
The collection
to define
S. (G)
action of
C (G)
l
( CL
e.l
(1) = 0
for used
E 0 10: G Ell. (0:) } l
is set-wise invariant under the
e.l
Less obvious is the fact that each
is an
algebraic integer linear combination of the irreiucible complex characters ,
e,
X~
and any such
is a complex linear combination of the orbital
characters
8. l
valued, then each
If all
e.l
lS
X~
e
are rational-
a true generalized
character and a rational class function. The functions
s(~:U)
are useful in estimat-
ing the size and de terming the structure of subgroups of prime power order. pr imi ti ve group of degree are all subgroups of
If
2p, and
@ with
u ~ rn0:
@
is a uni-
U <) \.i c C
\.i
~
for
59
some
a E 0
then
then
s (~: u)
> -32
>
s(~:U)
p
~
laul >
if
By using this fact,
P
2 ,
it is
possible to obtain far better estimates on the sizes of the Sylow q-subgroups of available classically.
than are
@
The lower bound on
s (lJl: U)
is obtained solely from knowledge of the "intersection numbers"
16. (a)
6. (p)
n
J
1
I ;
the author is
therefore quite confident that the method can be extended to cover more general situations. Some related results Surprisingly, the fact that
p
is prime is
not important for the proof of Theorem 1, and a better result may be formulated. Definition 3
Let
group of degree positive integer.
@
be a primitive rank
2p, Then
P
a @
3
(not necessarily prime) is of 2p-type provided
the degrees of the irreducible complex characters contained in the permutation character
e
are
1, p-l, P • Wielandt showed in [10] that any uniprimitive
FINITE GROUPS
60
group
0:
:::egree
2p
p
above de:inition
a prime, satisfies the
i.e., is a group of 2p-type.
By using the centralizer ring theory in conjunction with the "gap" theorem [11, 17.4] it is easy to show that all groups of 2p-type have degree 2p The
=
2
m
+ 1
for some odd positive integer
follo~ing
m.
result emerges from the proof of
Theorem 1. Theorem l'
If
is a group of 2p-type (p
@
necessarily a prime), and
m
prime
then
m
=
=
2p
2
m
+ 1
not
for some
3 •
The author has also proved a theorem conCerning groups of degree n*
3p
p
to be the set of primes
does not have the form Define
n**
a prime.
>
p
2 m + 12
such that
7
2 m + 15
or
to be the set of primes
which satisfy at least
f
A)
If
or
3p - 1 , then
B)
4
q
2
48p
.
p - 2 (mod 3 )
of A), B) , C) below:
~
is a prime divisor of q
Define
~
1 5
-
~
"4 3 p
is the exact power of
p - 1
+110
2
dividing
61
and if
p - 1
P - 1
of
(mod 3) C)
=
P
or or
f
q
2
3p q
1 + 2q
1 , then either
FP
1 5
~
is a prime divisor -
f
2
1
+ 110
1 + 4q
or
q
where
q
is
a prime. Theorem 2
If
P
is a prime in
uniprimitive group P E n* acter
and 8
@
@
on
3p
n* n n**
then no
letters exists.
If
exists, then the permutation char-
is a sum of irreducible complex characters
the characters distinct, real-valued,
are
and algebraically conjugate.
The proof uses the theory of characters of small degree [2], [3], the modular theory, of the centralizer ring.
and the theory
FINITE GROUPS
62
LITERATURE
1.
Brauer, R. "On groups whose order contains a pr ir..e to the fir st power," 1,11. Art:. J. Math. 64 (19"+2)
2.
Feit, ','1. "Groups with a cyclic Sylow subgroup." Nagoya Math. J. 27 (1966).
3.
Feit, W. "On finite linear groups." Algebra 5 (1967).
4.
Green, J.A. "A transfer theorem for modular representations." J. Algebra 1 (1964).
5.
Higman, D.G. "Finite permutation groups of rank 3." Math. Z. 86 (1964).
6.
Ito, N. "On uniprimitive groups of degree 2p." Math. Z. 78 (1962).
7.
Scott, L. "Uniprimitive groups of degree kp." Doctoral dissertation, Dept. of Math. Yale U. 1968.
8.
Sims, C.C. "Graphs and finite permutation groups." Math. Z. 97 (1967).
9.
Tamaschke, O. "A generalized character theory on finite groups." Proc. Internat. Conf. Theory of Groups, Austral. Nat. Univ., Canberra, August 1965, pp. 347-355. Gordon and Breed Science Publishers, Inc. 1967.
J.
10.
Wielandt, H. "Primitive Permutationsgruppen von Grad 2p." Math. Z. 63 (1956).
11.
Wielandt, H. Finite Permutation Groups. Academic Press 1964, New York.
SOME NEW SIMPLE GROUPS OF FINITE ORDER
Zvonimir Janko (Monash University, Melbourne)
We prove the following result: Theorem.
Let
G
be a non-abelian finite
simple group with the following properties: (i) T
of
G (ii)
The cent er
Z(T)
of a Sylow 2-subgroup
is cyclic. If
z
is the involution in
then the centralizer tension of a group
H E
of
z
of order
in 32
G by
Z(T) is an exAS •
We then have the following possibilities. G
has only one class of involutions, then
order
50,232,960
character table.
G
If has
and a uniquely determined If
G
has more than one class 63
64
FINITE GROUPS
of involution, then
G
has order
604,800
and a
uniquely determined character table.
The existence of a simple group of order 50,232,960
was shown by Graham Higman and J. McKay.
By a result of S. K. Wong follows that any simple group of that order must satisfy the conditions (i)
and (ii) of our theorem.
Also John G. Thompson
has shown that a simple group of that order must possess a subgroup isomorphic to
PSL(2, 16)
and
so using this fact the work of Graham Higman then shows that such a group is unique. The existence of a simple group of order 604,800 was shown at first by M. Hall, Jr.
Also a
very elegant geometric construction of this group was obtained by J. Tits.
It was proved by D. Wales
that there exists only one simple group of order 604,800.
ON JANKO'S SIMPLE GROUP OF ORDER
50,232,960
Graham Higman and J. McKay
1.
INTRODUCTION Z. Janko in [1] considered a finite simple
group
G (i)
such that G
contains an involution whose
centraliser is an extension of an extraspecial group of order
32
by
SL(2, 4)
In particular, he showed that if furthermore (ii) then
G
all involutions in has order
50,232,960
character table given in [1].
G
are conjugate
and has the He left open the
question of the existence and uniqueness of such a group.
Here we report briefly on work, partly
group-theoretic and partly computational, which shows that a group does exist satisfying (i), 65
(ii),
66
FINITE GROUPS
and also G
(iii)
has a sUbgroup
extension of
SL(2, 16)
morphism of order Conditions (i)
H
which is the
by an outer auto-
2.
to (iii) determine
isomorphism, and
G
G
up to
has an outer automorphism
extending the outer automorphism of
H
(i.e.
extending the automorphism of order
4
of
SL(2, 16)
2.
).
PERMUTATION CHARACTERS The first step was a systematic search,
carried out on the computer, for characters of
G
which might be permutation characters corresponding to large subgroups. character (a)
Necessary conditions for a
~
to be a permutation character are
~
contains the principal character just
once; (b)
~
(c)
~(x)
for all (d)
~
divides
(1)
(1)
IGI
is a non-negative rational integer x
in
G;
divides
hi ~ (x)
to a class containing
h. l
, if
x
belongs
elements; and
67
(e)
for all
all positive integers ~
By (c),
x
in
G
and
k. conjuga~e
contains algebraically
characters with the same multiplicity, so we add these together to obtain the rational table of to the of
G , with 14
G.
14
firs~
characte~
characters, corresponding
conjugate classes of cyclic subgroups
Then we generate the
2
13
combinations
of these characters which satisfy (a) all multiplicities are for conditions (b)
0
or
to (e).
and in which
1 , and test them
This takes
6
seconds.
Besides the principal character, there emerge characters of degrees
6156, 19380
and
20520,
corresponding, possibly, to subgroups of orders 8160, 2592 be subgroups
and H
2448.
The first of these could
as in (iii)
be subgroups isomorphic to
above; the last could PSL(2, 17)
Repeating the process, but allowing also multiplicity
2
takes
7
minutes.
Of the
characters that emerge, some can be seen to correspond to subgroups known to be present (if
G
exists at all), such as centralizers of involutions, and others can be eliminated.
But there are also
FINITE GROUPS
68
characters which might correspond to subgroups of whose presence we cannot be certain, in particular. subgroups isomorphic to
PSL(2, 19)
Finally, using a different program, assisted by the fact that
G
has only one non-principal
character of odd degree, the factors of
IGI
were
examined in increasing order of magnitude, with no bound on the multiplicities, except the natural one that the mUltiplicity of an absolutely irreducible character shall not exceed its degree. done up to degree
This was
17442; two new possible degrees
emerged, including
12312, corresponding,
presumably, to the
SL(2, 16)
subgroup of index
3.
2
in
contained as a
H .
GENERATORS AND RELATIONS At this stage, it seemed best to assume the
existence of the subgroup
H , and see what
consequences this would have. We denote elements of
GF(16)
by
i 2 w oj , where w + w + 1 = 0 , so that 2 and 0 + wo + 1 so that 0 5 = 1 0 of
SL(2, 16 )
are matrices
(: ~)
with
0
and
3 w = 1 , Elements a, b, c, d
69
in
GF(16)
and
ad - bc
<SL(2, 16) , u>
group is
(::
2 b 2)
d
<SL(2, 16), u 2 >
normaliser GL(2,
4)
,
L
in
V
u
Then
H
.
of
of order
G
4
where
(1 *1)
The elements abelian subgroup
1 ; and its automorphism
=
=
1
and
is
form an elementary
H
16, whose
must be a split extension by
acting naturally (from a consideration
of the structure of the centraliser of an involution). But
N (V) H
is only of order
480
Thus
G , if
it exists, can be generated by an amalgam where
H
U L ,
For reasons which will be
H" L
mentioned in a moment, it seems best to consider not only
G , but also an extension
G l
of
G
by
an outer automorphism extending the automorphism of H
induced by
amalgam L H
l l
u
HI \J L l
To this end, we form first an where
is the extension of f'\ L
l
=
HI
= <SL(2, 16), u>
V
by
fL(2, 4)
and
, with
N (V) Hl
It can be verified that this amalgam is generated by the relations
Hl
and an element
t , subject to
70 (A)
FINITE GROUPS t
2
= 1
t-1C :)t = t-lr W 2)
t-lut
t
u
e
~)
(w
t -1
(1
(1
:) t
w2 0 2) 1
2)
W
3
Further relations, which must hold in
G,
though not in the free product generated by the amalgam, follow by considering the normaliser of the cyclic group generated by
(w
w2 ) .
The
structure of this normaliser follows easily from
[1]; it is a split extension of , where elements in and other elements of normalisers of
»
w 2
«W
PSL(2, 9)
centralise
PGL(2, 9) in
by
H
invert and
L
can be identified with subgroups of this extension in essentially only one way, and this leads to the relations
71
[t (1 1)] [t(p4 P)t(p2 p7f
(B)
u
4
2
1 .
Using relations (A) and (B) it is fairly easy to see that the element
transforms one subgroup of
SL(2, 16)
of order
34
into another, and hence to find an element not in H
which normalises a cyclic subgroup of
order
17
H
of
Drawing once more on [1] for the
structure of a Sylow 17-normaliser, this gives another relation, which simplifies to
( C)
1.
Thus if SL(2, 16)
G u
2
implied by (A),
exists, it is generated by and
t , subject to the relations
(B) and (C), and possibly others.
However, a little experimentation suggests that the relations we have are already sufficient.
If
this is so, then, since the relations we have are implied by (i),
(ii) and (iii), there can be at
most one simple group
G
satisfying these conditions.
FINITE GROUPS
72
Furthermore, if such a group exists, any two embeddings in it of the amalgam equiv~lent
HV L
under its automorphism group.
are In
partir.ular, the automorphism induced by conjugation by
u
in
H \J L l l
extends to an automorphism of
G , necessarily outer, because otherwise the Sylow 17-normaliser of
4.
G
would be too large.
COSET ENUMERATION
To check that we have indeed enough relations, and that these relations do not imply total collapse, we return to the computer, and carry out an enumeration of the cosets of
H, using a program
based on methods described by J. Leech [2J.
For
this purpose, we have to replace the matrix notation for
SL(2, 16)
and relations.
used above by one using generators Since the critical factor in
attempting the enumeration is the size of the immediate access store available, and we store the effect on each coset of each generator and its inverse, it is advantageous to minimise the sum of the number of involutory generators and twice the number of non-involutory generators.
73 This part of the work was in fact done twice, simultaneously and independently, by one of us (J. McK.) at Chilton and by M. J. T. GUy at Cambridge, using different programs and slightly different relations.
In both cases the answer
produced was the hoped-for one, that cosets in
G c
s2
(ac) =
(sa)2
b 5 tb- 5 t
=
where t
-2
a, b, c
ab
-3
3
= (bc) 2
(sc)2
= sbsb- 4
(b-2ctb4ct) 2 b
has
6156
The relations used by Guy were
2
t2
H
=
=
abab
(at)2 (ct)4 s
-4
ab
(bt)3
=
b 2 tb- l abtb- 2 a
=
u
completed in 77 seconds.
=
= =
is as in section 3.
1
= (b 2 st)3
2 3 3 7 4 ctab ctb ab ctactb ab ct
generate
3
SL(2, 16)
s
1 , 2
,and
His enumeration was No attempt has been made
to tidy up these relations, and it is, in fact, known that some of them are redundant.
However, to
delete redundant relations might well increase the running time, and aggravate the danger of generating so many redundant cosets that one runs out of store. A further coset enumeration, this time on
FINITE GROUPS
74
cosets of the subgroup
t~e
, shows that
t~ese
two elements already generate
Jo
PROPERTIES OF
G
G
What the coset enumeration shows, of course, SL(2, 16)
t~at
:5
~e:ations
crder ~~G
(A),
u
subject to the
Evidently
Both
groups on
6156
G
l
letters.
~aving
suborbit lengths
2040,
and
2720 , and
suborbit of length length
1360
G
for
G.
G,
G
G
l
of
G
u
2
of order
are permutation is of rank
7 ,
1, 85, 120, 510, 680, G
2720
are primitive. subgroup of
~ormal
and
G l
SL(2, 16)
generate a normal subgroup
50,232,960
and
t
(B) and (C) generate a group
2 x 50,232,960. t
and
is of rank
8 , the
splitting into two of It follows that both So, if N
N
G
l
is a minimal
is transitive, and
since there is no characteristically simple group
0:
order
s~bgroup
relations u
2
6156, of
H
(B),
N (\ H Thus
(C)
and so
is a nontrivial normal N
contains
then show that N = G
That is,
SL(2, 16) N
contains G
, and t
is simple.
There are two classes of involutions in
H ,
75 255
(1 1),
conjugates of
order
32 , an d
68
with centralisers of
' conJugates
'h u 2 , Wlt
0 f
centralisers of order
By
conjugate in
, and hence to
G
to
is
(B)
In the permutation representation of degree U
2
has
76
f'lXe d pOln ' t s.
whose stabiliser is fixed by
If
H , and
a
Then
6g
=
and so is conjugate in
H
g
is another point
(including
a
, and for
itself) b
-1 2
u g
a
g
of
belongs to
either to
Suppose that for
2
6
is the point
u 2 , then there is an element
such that
u
a
6156,
u
2
H
or to
choices of
-1 2 g u g
G
6
is conjugate to
choices conjugate to
(1 1).
Then a + b
=
32 , we have
76
a
=
16,
b
=
60 ,
1920 That is, to be satisfied.
has the right order for (i) To show that it has the right
structure, we need only recall three facts.
First,
76
FINITE GROUPS
2 CH(u ) so
contains a subgroup isomorphic to
C (u 2 ) G
is not soluble.
Second,
contains a sUbgroup of order element of order on it.
3
~)
CL (( 1
)
normalised by an
which acts fixed-point-freely
Thus there is a composition factor of
2
on which SL(2, 4) '" So CG(u 2 ) so faithfully.
acts non-trivially, and
C r (u )
group of order by
16
SL(2, 4)
2
C (u) G
(B),
namely
32
t
(1 1)
abelian.
Hence
=
E·SL(2, 4) 2
normal in
CG(u)
E
Finally,
contains an element of order , so that G
E
a
8,
cannot be elementary
satisfies
(i), and hence also
(ii) . The computing described in this note was done on the Atlas computers at Chilton and Cambridge, which have a cycle time of done while one of us
2 f4
sec.
The work was
(J. McK.) held a Science
Research Council research fellowship at Chilton.
REFERENCES 1.
Z. Janko, Some new simple groups of finite order I
2.
(to appear) .
J. Leech, Coset enumeration, article in Computational Problems in Abstract Algebra,
77 Pergamon Press, 1968.
The Mathematical Institute 24-29 St Giles Oxford The Atlas Laboratory Chilton Berks
THE SIMPLE GROUP OF ORDER
604,800*
Marshall Hall, Jr. and David Wales
Z. Janko [3J has characterized a simple group G
in terms of the centralizer of an involution in
the center of a 2-Sylow group the center
Z
of
P
E
fully on
32
of order E
If
involutions then
G G
He assumes that
is cyclic and that the central-
izer of the involution in group
P.
by
Z
AS
is an extension of a
,
which acts fai th-
has two conjugate classes of has order
604,800
and a
uniquely determined character table. Proceeding independently of Janko, we have shown [2J the existence and uniqueness of a simple *This research was supported in part by ONR contract N00014-67-A00094-00l0. 79
FINITE GROUPS
80 group of order
604,800.
The degrees of the
characters in the principal 7-block,
B (7) , can O
be determined using techniques described in [1]. ~\
7-Sylow group is its own centralizer and its
normalizer is of order c~aracters
in
B (7) O
a 5-block of defect 1
From the degrees of
42
it follows that 1
G
possesses
and two 3-blocks of defect
The centralizers and normalizers of these
jefect groups can be determined from the order of G.
Then the degrees of the characters in these
blocks and a number of the conjugate classes in G
can be found.
It is then possible to see that
G
possesses a 2-block of defect
mentary Abelian defect group.
2
with an ele-
This provides two
more degrees of characters, and using the orthogonality relations, the character table can be completed.
The table is given at the end of this
paper. The next step is to show that s~bgroup
of order
3-Sylow group
P3
6048
G
contains a
The normalizer
is of order
216
and if
N
of a R
is
81
4
an element of order of order
N n C
and
96
in
N, then
C
is of order
= C(R)
is
These
24
three groups can be determined specifically from the character table.
By examining the multiplicity
of the trivial character in the restrictions it can be shown that the group generated by group. H
Call it
is of order
group
H
6048
Nand
C
is a proper sub-
It then is easy to see that and isomorphic to the unitary
D (3) 3
We can now show the existence and uniqueness of
G
as a permutation group on the
of
H
The permutation character
100 ~
cosets
correspond-
ing to the permutation representation can be found from the character table.
x=
~O
+ '1 + ' 7 ·
is of rank
It is
The permutation representation
3, and as a 7-element fixes exactly
two points, it readily follows that the stabilizer of a point,
G ,has orbit lengths a
1, 36 and 63 .
The permutation representation of orbit of length
36
H
on the
is on the cosets of a subgroup
FINITE GROUPS
82 of order
168.
Such a subgroup can be found and
is unique to within conjugation in the permutation character of its character on the
63
H
Knowing
on the
36 orbit,
orbit is determined,
and it follows that the tion of
H
H.
63 orbit is the representa-
on cosets of the centralizer of an
involution. It is now only necessary to show that this representation can be extended uniquely to a group of order
604,800 .
In
H
the normalizer
of a 7-Sylow group is of order (a, c>
(1)
a
7
and
H S
::::
d
:::: 1, c
-1
2 3 a , c :::: 1 .
ac
a, c
and two further elements
are given at the end of the paper.
(a, b>, d
2
::::
1, dcd
of the two letters
a group of order In
and
where
Permutations for b
21
G
::::
00
-1
and
and the stabilizer 01
is
S
::::
(a, d> ,
168
the normalizer
group is of order
C
Here
42
,
N (7) G
and so
G
of a 7-Sylow must contain an
83 involution
t
such that a
tat
( 2)
Furthermore
-1
G
=
tc
ct .
=
t> , as
H
must
be maximal. It remains to show that unique and that the group 604,800.
t
exists and is
has order
The latter was done with the help of
Peter Swinnerton-Dyer on the Titan computer at Cambridge University.
The simplicity of
G
follows from an easy argument using the simplicity of
H.
From the character table, 100
letters.
letters tc
= ct
by
c.
Hence
(00),
t
(01)
fixed by
tat
=
a
of the values in 2 4
a
the two further letters As
-1
2
tb a tb t determined.
fixes
(00)
t
must move all
must interchange the two and also, as (12,
fixed
(31)
this determines
the 7-cycles of a containing 16
t
(12) and (31)
t
on With
determined, we find that
and has
7
further values
This is sufficient to determine the
84
FINITE GROUPS
permutation completely as an element of
H
and in
fact 363 a b ada .
(3) ?elations ~:~e
t~at
t
( 2)
and
(3) are now enough to deter-
completely.
The computer verified
the subgroup of
exactly g~oup
H
and so
of order
G
=
fixing
604,800
00
is
is the desired
and is unique.
Suzuki has also constructed the group as a subgroup of index
2
a certain graph with ~as
over
of the automorphism group of 100
points.
Jacques Tits
a similar construction using certain geometries GF(4)
Recently John Lindsey has found the
matrices for a 6-dimensional projective representation of
G
over the complex numbers.
85
element
e
a
J
R
K
TJ
TR
order
1
7
2
4
8
6
12
C (X)
g
7 1920 96
8
24
12
~O
1
1
1
1
1
1
1
~1
36
1
4
4
0
1
1
~2
90
-1
la
-2
0
1
1
~3
160
-1
0
0
0
0
0
~4
225
1 -15
-3
-1
0
0
~5
288
1
0
0
0
0
0
~6
300
-1 -20
4
0
1
1
~7
63
0
15
3
1
0
0
~8
126
0
14
2
0
-1
-1
~9
70
0 -la
2
0
-1
-1
~10
70
0 -la
2
0
-1
-1
~ll
175
0
15
-1
-1
3
-1
~12
224
0
0
0
0
0
0
~13
224
0
0
0
0
0
0
~14
14
0
-2
2
0
1
-1
~15
14
0
-2
2
0
1
-1
~16
21
0
5
1
-1
-1
1
~17
21
0
5
1
-1
-1
1
336
0
16
0
0
-2
0
~19
189
0
-3
-3
1
0
0
~20
189
0
-3
-3
1
0
0
*18
FINITE GROUPS
86
element
TIJ
order
10
C(X)
20
1
1
1
1
1
1
1
-1
-1
o
o
o
o
,2
0
0
6
o
1
1
,
0
0
4
1
-1
-1
0
0
5
-1
o
o
0
0
4
1
-1
-1
0
0
o
o
o
o
0
0
-1
-1
-1
-1
-1
-1
6
o
1
1
-2
1
'" 0 '1
~ ~
.et '5 'il 'f
6 7
~8
'il
9
L o
0
-2
1
0
0
-5
1
o
o
~12
0
0
-4
-1
1
1
Vu
0
0
-4
-1
1
1
'+'14
6
2
-1
-6
V1S
6
2
-1
-6
'+'16
0
0
-3
0
-6
'+ 17
0
0
-3
0
-6
'18
o
o
'19
o
'20
o
'+ 10
Vl l
2 1
6 6
1 2
1 2 2 1
-6
-6
2 1
-6 -6
o
1 2
87 TI
TI
3
5
5
36
300
element
T
T
order
3 1080
C(X)
1
2
TI
rr
1
2 1
,...2
TIT
.. T
5
5
15
15
300
50
50
15
15
~O
1
1
1
1
1
1
1
1
~1
9
0
-4
-5
1
1
-1
-1
~2
9
0
5
5
0
0
-1
-1
~3
16
1
-5
-5
0
0
1
1
~4
0
3
0
0
0
0
0
0
~5
0
-3
3
3
-2
-2
0
0
~6
-15
0
0
0
0
0
0
0
~7
0
3
3
3
-2
-2
0
0
~
9
0
1
1
1
1
1
1
~9
7
1
58
0
0
-8
~10
7
1
58
0
0
-8
'~11
-5
1
o
o
o
o
o
~12
8
-1
1-48
8
8
~13
8
-1
1-48
~14
5
-1
38
~15
5
-1
38
~16
3
0
3+ 8
~17
3
0
3+8
~18
-6
0
-4
~19
0
0
38
'1J
0
0
38
8
20
58
1
58
2
2
1
o 2 1
1-48 1-48 38
1
38
2 1 2
1 2
1+8
1
3+8
-4
-28 1+8
2
3+8
-28
2 1
28 28
1 2 1 2 1 2
1
2 1
1+8 1+8
-28 -26 1+8 1+6 26 26
2 1 2 1 2 1
1
1 2
1+6
1+e
2
1
1 2
1
8
2
-8
8
0
0
0
0
8
1
8
2
2
-8
1
2 1
8 8
2 1
-1
-1
o
o
o
o
FINITE GROUPS
88
Permutations a =
(00) (01) (02,03,04,05,06,07,08) (09,10,11,12,13,14,15) (16,17,18,19,20,21,22) (23,24,25,26,27,28,29) (30,31,32,33,34,35,36) (37,38,39,40,41,42,43) (44,45,46,47,48,49,50) (51,52,53,54,55,56,57) (58,59,60,61,62,63,64) (65,66,67,68,69,70,71) (72,73,74,75,76,77,78) (79,80,81,82,83,84,85) (86, 87, 88, 89, 9 0, 91, 92) (93, 94, 95, 96, 97, 98, 99)
b
(00) (01,02,09,16,23,20,30,17) (03,35,13,29,31,24,25,11) (04,26,27,07) (05,21,14,10) (06,19,32,36) (08,18,28,22,15,12,33,34) (37) (38,92,67,77,99,89,49,84) (39,59,82,46,88,54,52,68) (40,55,47,81,75,95,61,78) (41,44,63,58,72,65,50,51) (42,76,53,93,86,80,79,57) (43,85,48,70,96,83,66,94) (45,97) (56,87) (60,71,91,69,90,73,64,74) (62,98)
c = (00) (01) (02,28,21) (03,23,18) (04,25,22) (05,27,19) (06,29,16) (07,24,20)(08,26,17) (09,13,14) (10,15,11) (12) (30,36,34) (31) (32,33,35) (37,50,63) (38,45,60) (39,47,64) (40,49,61) (41,44,58) (42,46,62) ( 43, 48 , 59) (51, 65 , 72) (52, 67, 76) (53, 69, 73) (54,71,77) (55,66,74) (56,68,78) (57,70,75) (79,89,96) (80,91,93) (81,86,97) (82,88,94) (83,90,98) (84,92,95) (85,87,99)
89
d
=
(00) (01) (02) (03,05) (04,25) (06) (07,20) (08,17) (16,29) (18,27) (19,23) (21,28) (22) (24) (26) (09,14) (10) (11,15) (12) (13) (30) (31) (32,35) (33) (34,36) (37,44) (38,88) (39,64) (40,90) (41,50) (42,95) (43,97) (45,82) (46,92) (47) (48,86) (49,83) (79,89) (58,63) (59,81) (60,94) (61,98) (62,84) (80,85) (87,93) (91,99) (96) (51,72) (52) (53,68) (54,71) (55,66) (56,69) (57) (65) (67,76) (70,75) (73,78) (74) (77)
t
=
(00,01) (02,74) (03,73) (04,72) (05,78) (06,77) ( 07 , 7 6) (08, 7 5) (0 9 , 34) (1 0 , 33) (11, 3 2) (1 2 , 31 ) (13,30) (14,36) (15,35) (16,71) (17,70) (18,69) (19,68) (20,67) (21,66) (22,65) (23,53) (24,52) (25,51) (26,57) (27,56) (28,55) (29,54) (37,91) (38,90) (39,89) (40,88) (41,87) (42,86) (43,92) (44,99) (45,98) (46,97) (47,96) (48,95) (49,94) (50,93) (58,85) (59,84) (60,83) (61,82) (62,81) (63,80) (64,79)
90
FINITE GROUPS
Bibliography :lj
Hall, M. Jr.:
A search for simple groups of
orders less than one million.
:2.
Hall, M. Jr. and D. Wales: of order
3.
604,800.
To appear. The simple group
To appear.
Janko, z.:
Some new simple groups of finite
order, I .
To appear.
LE GROUPE DE JANKO D'ORDRE 604.800. J. Tits
Soit groupe
H
l'hexagone generalise associe au
G (2) 2
points de
H
On sait [3] que l'ensemble des
points d'une hyperquadrique sur
E2.
Un plan de
Q
contient aucune droite de nocmaux distincts toute droite de une droite de est associe
a
Soient
a
peut etre identifie
p, p' Q
H
Q
l'ensemble des
de dimension
est dit normal s'il ne H; deux plans sont dits associes si
rencontrant
p
et
p'
A
un et un seul autre plan normal. H'
=
est
On montre que tout plan normal
l'ensemble des droites de
l'ensemble des paires de plans associes, point et
5
H' UP U {oo}.
la relation binaire symetrique
91
Dans
00
H, P un
A, considerons
I C A
x
A
definie
?2
FINITE GROUPS
20mme suit:
pour
x)
( 00,
I
E
(d, d')
E
d, d' E H'
<=>
I
~
X
E
et
P
les droites
d
et
d'
sont
disjointes et non opposees dans
(i.e. il existe une
H
et une seule droite de rencontrant ~
Soit
I.
elements de
J
automorphismes (resp.
(resp. de
n
p.1
et
K'
Soit
intersections
Pi
une droite de
Q.
K
d'
);
'1.0
le groupe des permutations de
J
conservant
J'
d
d
H
n
est
qj
A
le groupe des
induits de fa90n evidente par les ("Collineations") de
K'
H
et soit
le groupe des elements de
J
) qui SOnt des permutations paires
A.
THEOREME.
On a
(J:J' )
(K: K' )
2.
Le groupe
93
J'
est un groupe transitif de permutations de
l'ensemble d'ordre J
a
(resp.
J'
elements
604.800.
A
i
il est simple,
Le sous-groupe des elements de
laissant fixe le point
J'
sous-groupe de
100
K
(resp.
K
I
est le
00
Tout automorphisme
).
s'etend de maniere unique en un automor-
phisme de
J , et tout automorphisme de
interieur, de sorte que
~
Aut (J')
J
Jest
.
Un etape essentielle de la demonstration consiste evidemment
a
montrer l'existence d'une
permutation impaire de
A, appartenant
ne conservant pas
Elle ressort du
00.
LEMME.
a
J
et
11 existe une
permutation involutive
a
proprietes suivantes, ou
de K
A
possedant les
= (ql' q2)
E
P
et
d E H'
(i)
a (00)
=
a(TI)
TI
00
(ii)
si
( TI ,
K)
E
I
,
alors
( iii)
si
( TI ,
K)
~
I
,
alors, apres permu-
a (K)
=
K
tation eventuelle de ql
et
q2
, les
94
FINITE GROUPS intersections
Pi
n qi
sont des points, joints par une droite et (iv)
si
(TI,
d)
a(K)
E l , a(d)
=
d'
est l'unique paire
d
(v)
si
(TI,
d)
~
I
d' E H',
n
p.
1
;
, il existe une seule droi te
dIE
rencontrant d , et
a(d)
H'
Pl' P2
et
est
l'unique droite
E HI
distincte de
et
d
d',
et contenant le point d
L'involution appartient
a
n
d'
a , qui est manifestement unique, J
Les demonstrations sont assez longues, parce G~e ~ais
requerant de nombreuses distinctions de cas, elementaires. Les resultats precedents ont ete suggeres par
95
la lecture de [1].
La representation, decrite
ici, du groupe de Janko comme groupe de permutations de degre
100
est bien entendu equivalente
a celle obtenue par M. Hall au moyen d'un ordinateur; la connaissance de celle-ci a aussi oriente nOs recherches.
REFERENCES 1.
D. Higman and C. Sims, A simple group of order
2.
44.352.000.
To appear.
Z. Janko, Some new simple groups of finite order, I.
To appear in the Proceedings of a
Conference on Group Theory (Rome, 1967). 3.
J. Tits, Sur la trialite et certains groupes qui s'en deduisent, Publ. Math. I.H.E.S., 2
(1959), 13-60.
Bonn, Decembre 1967
LINEAR GROUPS OF DEGREE
6
AND THE
HALL-JANKO GROUP John H. Lindsey 11
Finite groups
G
having a faithful,
irreducible, quasi-primitive, unimodular representation of degree
6, over the complex field, have
essentially been classified. extension by
is a central
AS' A , A , L (7), L 2 (11), L 2 (13), 6 7 2
L (4), U (3), U (3), 0 (3), 3 4 3 5 group of order
G
or the Hall-Janko
604,800, or
over one of these groups.
G
is of index
2
The only questions left
open are the automorphism groups of the central extensions by
A 6
and
L (4) 3
of the central extension of representation of degree
and the uniqueness 2
6
by
L (4) 3
with a
6
In the course of classifying groups of degree
6, the following generalization of Feit's
theorem was proved: 97
98
FINITE GROUPS
Let h = g
TI
G
,
be a group of order
~
TI
{primes
> r
abelian sUbgroup of
G
- l}
2
either A.
or
B.
, and
of order
X
a faithful representation not a power of
glh
h.
g H
where is an
Let
of degree
G
have
r - 1
over the complex numbers.
Then
G
has a normal subgroup of order
h
or
P
hip,
a prime number,
There exists a subgroup of index
1
or
2
with
GO
of
G
GO
isomorphic to a central extension by
L 2 ( r)
A group of index of
Z6
by
U (3) 4
matrices in
(1)
2
~(w»
over a central extension
where
w
is a primitive
1.
All diagonal matrices of order
1 ,
determinant ( 2)
L 2 (r)
was generated by the following
GL(6,
third root of
x
All permutation matrices, and
(3 )
1 w - w
1
1
1
1
w
w
1
w
w
0
0 -1
-1
-1
-1 -1
-w
-w
-w
-w
3
and
99
It is shown that these generate a finite group by examining the
126
conjugates of a
permutation matrix corresponding to a transposition. 45
conjugates are monomial.
conjugate to order
16 -
3 , where
entries are
(1/3)M M
81
by diagonal matrices of
is a matrix all of whose
1
There is a unique group of order
conjugates are
2
and
Gl/Z
G l
with center
Z
G , the Hall-Janko group
~
satisfying the following: (A)
The Sylow 7-subgroup of
G
normalized by an element of order (B)
is
.
4
The inverse image, under the homo-
morphism
G l
of order
6048
->-
of a subgroup
Gl/Z
is isomorphic to
of
U (3) 3 Z
x
G
U (3) 3
G l
has two conjugate, faithful, irreducible representations of degree
6
in the complex field.
representations can be written in character table of
Gl
~(15,
can be given.
These
17 ).
The degrees
of the faithful irreducible representations of are
The
G l
6, 6, 64, 64, 50, 50, 216, 14, 84, 126, 126,
252, 56, 56, 448, 350 Existence of (mod 3)
G l
and
336.
was verified by taking
a representation of degree
6.
This
FINITE GROUPS
100 ~odular ~as
a
~hen,
representation restricted to 3
Z x U (3) 3
dimensional invariant subspace,
under
G l
generators of
, G
l
V
has
100
V.
images, which the
permute exactly as their images
permute the letters described in the Hall-Wales paper. Unique unitary matrices over the complex :ield were obtained for a
6
dimensional
representation of generators of ~ormalizer ~ormal
G
l
after the
of a Sylow 7-subgroup was written in a
form.
ON THE ISOMORPHISM OF TWO GROUPS OF ORDER
44,352,000
Charles C. Sims
1.
INTRODUCTION At the Group Theory Symposium held in Urbana
November 24, 1967 G. Higman described a simple doubly transitive permutation group of degree
176
with the same order and character table as the simple group discovered by D. G. Higman and the author.
In this note a proof of the isomorphism
of these two groups will be sketched.
The proof is
computational in the extreme and was carried out entirely on a computer.
2.
THE GROUP OF G. HIGMAN G. Higman defined his group as the auto-
morphism group of a "geometry" whose objects are 101
102
FINITE GROUPS
points and subsets of points called quadrics such that (i)
There are
(ii)
176
points and
Each quadric contains
each point is in (iii)
50
176
50
quadrics.
points and
quadrics.
Any two distinct points are in
14
quadrics and any two distinct quadrics have
14
points in common. (iv)
There is a polarity interchanging points
and guadrics, preserving incidence. Higman's definition of the geometry is equivalent to the following:
Let
A subgroup of the symmetric group be called a
PGL(2, 5)
orbits of length C
of
168
1, 1
B
{1,2, ... ,8}
=
S8
on
if it has order and
PGL(2, 5) 's
6
S8
B 120
will and
has one class
The set of points in
the geometry is P
~he
B U C •
quadrics are in
~oints.
If
B U {H I~
H
€
1-1
correspondence with the
b
€
B , then the quadric
I
H
€
C , then
C, H
b
H
=
L(b)
is
b}
has orbits on
C
of length
103 1, 5, 12, 30,
60
and
The quadric
60
contains the fixed points of orbits of length Let
Q
1, 5, 12
H
on
B
and
30
of
denote the set of quadrics.
morphism group of the geometry of order
44,352,000.
is isomorphic to PSU(3, 52)
L(H) and the H
on
C.
The auto-
(P, Q)
is simple
The stabilizer of a point
P2:U(3, 52)
, the extension of
by a field automorphism.
Higman distinguished certain other subsets of points called conics. ~ich
points are
1100
These are sets of
are contained in
conics.
B
8
quadrics.
8
There
is a conic and the only
automorphism of the geometry fixing a conic point-wise is the identity.
3.
THE GROUP OF HIGMAN-SIMS Let
Sand
B
denote the sets of points
and blocks, respectively, in a fixed Steiner system S(3, 6, 22) (i)
(ii) of
Thus
Is[:= 22. B
is a set of
77
6-element subsets
S . (iii)
Any three distinct points are contained
104
FINITE GROUPS
in a unique block. Define a graph
*
where
~
with vertex set
is a new symbol.
points in
Connect
*
to the
S , connect each point to the blocks
containing it and connect two blocks if they are disjoint.
The automorphism group of
~
88,704,000
and contains a simple group
index
The stabilizer of a point in
2.
isomorphic to
has order G
of
G
is
M . 22
THE ISOMORPHISM
4.
Let us denote the group defined in section 2 by
GH
and the group defined in section 3 by
HS
To prove that these two groups are isomorphic it is enough to show that isomorphic to
M 22
character table of
GH
contains a subgroup
By inspection of the M 22
possible degree less than
we see that the only 100
permutation representations of 56
and
of an
77
Thus representing
of transitive M 22 GH
are
1, 22,
on the cosets
M , we must get a primitive rank 22
3
group
105 of degree
100
with subdegrees
1, 22
and
77.
It is not hard to show that this group would have to be an automorphism group of the graph defined in section 3. To show that
GH
contains an
construct a geometry on which
M 22
M 22
we
acts and prove
that this geometry is isomorphic to the geometry defined in section 2.
M 22
subgroups isomorphic to
has two classes of
A
There are
7
176
subgroups in each class and the two classes are interchanged by an outer automorphism of
M 22
A sUbgroup in one class has orbits of length 35
and
126
on the other class.
15,
We define a
new geometry in which the points are the sUbgroups in one class of
A 's 7
and the quadrics are the
subgroups in the other class. quadric
K
A point
and a
H
are incident if
IH n KI
72
or
168.
We must now show that the geometry
(P, Q)
of section 2 is isomorphic to the geometry (pI,
Q')
just constructed.
that we turn to the computer.
It is at this point The two geometries
were explicitly determined and a particular
106
FINITE GROUPS
isomorphism exhibited.
The construction of the
geometries is relatively easy and it is sufficient to say that programs were written to compute the two
176
176
x
incidence matrices of the
geometries. Determining an isomorphism was somewhat more difficult.
G. Higman showed that the
point-wise stabilizer of a conic
B
in
(P, Q)
is trivial by showing that there is a "natural" identification of the points in set of
PGL(2, 5) 's
P - B
with the
in the symmetric group on
This identification is obtained as follows: i,j
E
B,
i
x
in
P - B
8
quadrics.
t
j
,
let
such that
C..
1.J
{i,
k t that
i, j
{i, y, z}
of order
j, x}
is contained in
y
(i,
z
in
C
such
jk 3
Given an element
5
in the symmetric group on
(i, b r )
(i, b ), l
j)
in
is contained in more than
composition of the maps and
be the set of points
, with the unique point
quadrics.
For
Also, define a map denoted by
which interchanges the point
B
B , the
(j, b ),
defines a permutation
r
f
r
on
107 The product
has a unique fixed point with the unique
x
group on g
B
x
PGL(2, 5)
which fixes
c l..]
on
i
Identify
in the symmetric and
j
and contains
•
A computer program was written to look for conics and to construct this identification.
The
only input to the program was the incidence matrix of
(P, Q)
The program was then given the
incidence matrix of
(P I,
Q')
instead.
If the
geometries were isomorphic, then a conic would be found and the identification determined.
If the
geometries were non-isomorphic, there were several possible results: (1)
No conic would be found.
(2)
A conic
identification of PGL(2, 5) 's
B'
would be found but the
P' - B '
with the set of
in the symmetric group on
G'
could
not be carried out. (3)
A conic
B'
and the identification
would be found. As it turned out the computer found a conic
108
FINITE GROUPS and constructed an identification of
3'
·.·:i th the set of
group on in
PGL (2, 5)' s
B'
(P, Q)
~
A map was chosen.
of the synunetric groups On
B'
onto a conic
pI
verified that
onto ~
P.
B'
B
induced an isomorphism B'
and
identifications, one could extend a map of
-
in the synunetric
of ~
pI
using the
B ~
uniquely to
A separate program
took quadrics to quadrics and
thus was an isomorphism.
5.
PW(3,
52) 'S
IN
HS
We close with a construction of the graph in section 3 which shows that subgroups isomorphic to has three classes of
HS
contains
PZU(3, 52)
A ' s. 7
PSU(3, 52)
Take the vertices of
the graph to be the subgroups in two of these classes. (a) H
n
KI = (b)
H
n
KI =
Connect two vertices Hand
Hand
K
K
are conjugate and
K
are not conjugate and
360
Hand 168
Rutgers University
if
A SIMPLE GROUP OF ORDER
898,128,000
Jack McLaughlin
Let
U
denote the unimodular group of a
polarity of unitary type on projective space of dimension group
3
over the field of
U4 (3)
9
elements (the ~
in Artin's notation), and let
denote the class of totally singular line stabilizers in
U.
primitive, of rank
The action of
3 , with subdegrees
If we ask for a primitive rank which a point stabilizer is orbits for
U
is
on Y
U
U
is
1, 30, 81.
3
groups in
and one of the
Y, the conditions worked out
by Donald Higman in [2] tell us that
162
is a
possible value for the other non-trivial orbit length. E. M. Hartley in [1] showed that
U
contains
the unimodular group on the projective plane of 109
110
FINITE GROUPS
order
4
(the group
index
162).
Let
isomorphic to
L (4) ) as a subgroup (of 3 be a class of subgroups
Y
L (4) 3
in
U.
One can verify the
following: (1) and
U
has
orbits on Y
2
x
Y , say
0'
0" . The action of
(2) or rank
U
on
3 , with subdegrees
Y
is primitive,
1, 56, 105 .
Following Higman-Sims [3] we make a graph
* U YuY.
whose vertex set is member of
Y.
Join
S
E:
Y
of the S-orbit of length L
E:
Y
with
(S, L)
E:
0'
to
30 .
On Join
members of the L-orbit of length to the
S
E:
Y
with
(S, L)
E:
*
Join
to each
* , the members Y, and the L 56
0'
has a transitive automorphism group. of the title is a subgroup of index
E:
Y on
to the
Y, and
The graph
!§
The group 2.
REFERENCES 1.
Hartley, E. M., Two maximal subgroups of a collineation group in five dimensions, Proc. Carob. phil. Soc. 46 (1950), 555-569.
111
2.
Higman, Dona1d G., Finite permutation groups of rank
3.
3 , Math. Z. 86 (1964), 145-156.
Higman, Dona1d G. and Charles C. Sims, A simple group of order
44,352,000,
Math. Z.
105(1968) 110-113.
The University of Michigan
A SIMPLE GROUP OF ORDER
448,345,497,600
Michio Suzuki
We construct a simple group
S
of order
448,345,497,600 . Comparison of order shows that
S
is a new simple
group not found in the list of simple groups published so far.
What we have proved is the
following theorem.
THEOREM 1. extension E 2 (4)
There exists a primitive transitive S
of degree
1782
of the simple group
, the Chevalley group of type
G 2
over the
field of four elements. Since
E (4) 2
is simple and there is no
characteristically simple group of order 113
1782,
FINITE GROUPS
114 Theorem 1 yields
COROLLARY.
S
is a simple group.
Thus the group group on rank
3
1782
S
is defined as a permutation
elements and as such
S
is of
in the sense of D. G. Higman [3].
The
parameters are k
416 ,
1365,
in the notation of [3].
A
100
and
~
The permutation group
defines a strongly regular graph
r
with
96
S
1782
vertices, such that the group of automorphisms of 1
contains
S
The proof of Theorem 1 depends on
the construction of a graph which turns out to be
r The construction of the graph depends On various properties of the simple group
E (4) 2
Among them the following result is crucial.
THEOREM 2.
The simple group
E (4) 2
has a
primitive permutation representation of degree such that it is of rank
3
416
and the stabilizer of
a point is isomorphic to the simple group of order
115 604,800
recently constructed by M. Hall [2].
Let
6
be the graph associated with the
permutation representation of
=
Let
E (4) 2
be the set of sUbgroups of
in Theorem 2. E 2 (4)
which
are conjugate to the center of a Sylow 2-group of E (4) 2
r
We construct a graph
r
vertices of we denote
as follows.
The
are a distinguished point, which
(00)
, vertices of
6
and elements of
=: r
(00) U 6 U
r
The edges of (a) 6
The point but none of
(b)
r
=
are defined as follows: (00)
is joined to every point of
=;
Two vertices of
6
are joined by an edge in
if and only if they are joined by an edge of
(c)
If
a s 6
and
b s
=,
a
and
b
are joined
by an edge if and only if the stabilizer of E 2 (4)
a
in
contains a non-identity element of the
subgroup
b
(d)
u
If
we join [u, v]
L
~
and
v v
are two subgroups of
u
and
1
but there is a subgroup
=,
then
by an edge if and only if w
of
=
such
116
FINITE GROUPS
that
=
(u, w]
=
(v, w]
1.
Here
the commutator subgroup of It is proved that
u
and
Aut f
denotes
(u, v]
v.
is transitive on
the set of vertices and contains the simple group S
of index
2
which satisfies the condition of
Theorem 1. The proof of Theorem 2 is in turn reduced to a similar proposition about the Hall-Janko group. The graph
6
is constructed from the representation
of the Hall-Janko group of degree
100
In fact
we may begin the construction starting from the trivial graph
f
consisting of four vertices and
l
no connecting edge.
The group
the symmetric group
S4
from
6
f
G 2
14
=
before.
vertices.
Aut f
2
G 2
construction the group It is the stabilizer of on G l closure of
Ll
Let
We construct
S4
Since
in
9 ,
It is not hard to verify
~
PGL(2, 7)
G2
By
is a subgroup of
G l ( co)
G 2
in the natural action
We may define
f
2 Ll
is
l
is transitive on the set of
vertices and in fact
of
f
in a similar way as we constructed
L
and
contains that
2
= Aut
on four letters.
be the set of involutions of the graph
G l
The set
as the normal
L2 L2
consists
f
117 of
21
involutions of
define
f
, G , =3 3
3
PSL(2, 7)
similarly.
We can
Continuing this
process we obtain 36
If 4 1
100
Ifsl
Here
,
If 3 1
HJ
=
=
G 3 G 4
,
416
G S
E 2 (2) Aut HJ ;
=
Aut E (4) 2
denotes the simple group of Hall-Janko.
The graph
f
is the graph
S
6
defined just after
Theorem 2. The proof of a crucial point, the group
G.
acting transitively on the set of vertices of
f.
1
1
depends on various properties, among which we single out two. Let
x
be a vertex of
stabilizer of to L 1- 1
G i l
x
in
G.
f.
1
Then
1
involutions of outside
H
and this isomorphism sends Thus, in the embedding of
.
and
=.1- 1
H
be the
is isomorphic =.
G _ i l
1
n H onto into
G.
1
do not fuse to involutions
=.1- 1 .
The second, more important, property is the following. of
x.
Let
As before let i
<
4 .
If
H
denote the stabilizer
u
is an involution of
118
Li
FINITE GROUPS which is not contained in Hn L
unique involution in u.
H , then there is a which commutes with
i
This proposition fails for
G S
reason that the sequence stops at construction of the simple group different.
This is the i
S
5
and the
is slightly
The proof of Theorem 1 uses a similar
proposition involving the subgroups of
L
Details of arguments are elementary but tedious, and will appear elsewhere. that the identification of
G4
We mention
uses the recent
work of M. Hall and Wales [2], and the isomorphism is proved by applying the recent characterization of The graph
f
E 2 (q)
by G. Thomas [4].
(and its dual)
is a strongly
regular graph which is pseudo-geometric but not geometric.
The graph
f
4
is even a pseudo-net
graph but is not a net or the dual of a net. -4
Thus
provides a counterexample to a conjecture of
Bruck [1].
A simpler counter-example to Bruck's
20njecture has been given by Bose. This work was inspired by the recent work of ... Hall, D. Higman and C. Sims. =a~e
The main idea
from the work of Higman and Sims on their
siDple group.
The author is grateful to them for
119 communicating to him their work before publication.
REFERENCES 1.
R. H. Bruck, Finite nets, 11.
Uniqueness
and imbedding, Pac. J. Math., 13 (1963), 421-457. 2.
M. Hall and D. Wales, The simple group of order
3.
(to appear).
D. G. Higman, Finite permutation groups of rank
4.
604,800
3 , Ma th. Z. 86 ( 1964), 14 5-156 .
Gomer C. Thomas, A characterization of the Chevalley group
G (q) 2
, Thesis, University
of Illinois (1968).
University of Illinois
SOME SIMPLE GROUPS RELATED TO . Dleter Hel d
M 24
l
The following characterization of the Mathieu-Simple-Group
THEOREM.
Let
group of order
EO 16.
of an involution of If
G
has been obtained:
be an elementary abelian Denote by EO
HO
the centralizer
in the holomorph of
EO .
is a finite simple group which possesses an
involution G
M24
z
such that the centralizer of
is isomorphic to
z
in
HO ' then only the following
possibilities occur:
1.
(I)
G
is isomorphic to
L (2) S
(Il )
G
is isomorphic to
M 24
(Ill)
G
has order
, or
210'33.s2'73·17
This work has been done under a DFG contract. 121
122
FINITE GROUPS
SOME PROPERTIES OF A GROUP (1)
The group
G
G
OF CASE III
possesses precisely two
classes of involutions with the representatives and
z
t. (2)
The centralizer
isomorphic to
HO ,and
Sylow 2-subgroup of
H
of
z
in
is
is the center of a ~2(H)
Moreover,
G.
G
is the
central product of three dihedral groups of order 8
to
H/~2
with amalgamated centers, L
2
(7)
,
( 3)
and
H
possesses a subgroup
is isomorphic
~2(H)
splits over
The centralizer
(H)
H*
of
t
in
of index
2
which
G
contains a normal elementary abelian subgroup of order
of
4
isomorphic to
H* . L (4) 3
sUbgroup isomorphic to (4)
H*/V
The factor group , and
H*/V
contains a
G
is an
extension of a non-abelian group of order
(5)
3
27
by a dihedral group of order
A Sylow 5-subgroup
abelian of order
52.
F
8
and 8
is elementary
The factor group
~(F)/F
is the central product of a quaternion group of order
is
1
Ss
A Sylow 3-normalizer of
of exponent
V
Q
extended by an automorphism of order
123 3
and a cyclic group
IQ
n zl = (6)
2
=
Z
and
Z
of order
~(S)/S
Further,
7
3
S
of
7
3
and
has precisely
conjugate elements.
S3 G
is a
2 3 .17
Frobenius-group of order G
is
is isomorphic to the direct
A Sylow 17-normalizer of
(8)
G
and of exponent
product of a group of order (7)
such that
~(~(F)/F)
A Sylow 7-subgroup
non-abelian of order
4
33
classes of
The centralizers of the
representatives of these classes can be obtained uniquely by the above information.
In particular,
G
has precisely two classes of elements of order
3
with representatives
C
H
E
and
precisely one class of elements of order the representative
EH,
(j
computes
-1
S
~(c)
7
with the representatives H* ,
E
'"
x
s
-1
L (7) 2
One f(cl)/
2.7 2
C(w)
'" <w>
x
AS ,
If ((j) I
=
C (s)
'" <s>
x
L (7) 2
, and
If (w) I
Remark.
with
w , and five classes of
elements of order (j
S
3.7 3
The proof leading to the above results
will be published in a forthcoming paper.
In this
FINITE GROUPS
:1.24 ~aper,
the existence problem will be attacked.
particular, the character table of
G
will be
computed. In
t~e
proof of the above theorem a new
group-orcer formula of J. G. Thompson plays a crucia::' :-::le.
In
AN ANALYSIS OF GROUP REPRESENTATIONS S. B. Conlon
Let
G
be a finite group and
tative ring with identity.
Let
Jt
R
a commu-
denote the
category of finitely generated left RG-modules. The representation ring is formed from integral
a (RG) (~
of the isomorphism classes subject to the relations ever
M "" M' Gl M"
(algebra
A (RG) )
-linear) combinations {M}
{M}
for
= {M'}
M
in
.1,
+ {M"}
when-
and in which mUltiplication is
given by the "tensor product representation," i.e. {M}·{M'} ~heorem
{M ~ M' }
holds in
If the Krull-Schmidt
Jt, then we have a natural
imbedding of the isomorphism classes in
a(RG)
and of
a(RG)
in
of Jt
{M}
A(RG)
In the classical case with
R
~
we know
that character values separate not only the 125
126
FINITE GROUPS
isomorphism classes of
~,
but, if extended
linearly, also separate the elements of
A(RG)
One can ask the same question for general When
R
is a complete
~_ocal
the Krull-schmidt theorem holds in
R.
noetherian ring ~
and Green's
work on vertices and sources and transfer goes through.
In [1]
it is shown that, provided one
has an adequate knowledge of the possible (absolutely indecomposable) sources that can arise, the elements of
A(RG)
can be systematically
separated and the question is really one of the sources. The reduction involves two ideas.
First, a
close look is taken at the decomposition of an G L
induced module source of vertex
RG 0
RP
L ,where
L
is a
P , by means of rings of endo-
morphisms and the notions of ideals and radicals in categories.
The analysis elucidates Green's
transfer theorem. The second idea is that of a canonical decomposition
A(RG)
into two-sided ideals
~
Ap(RG)
Ap(RG)
, of
, where
~
complete set of non-conjugate p-subgroups G
and where
p
A(RG) is a P
is the characteristic of the
of
127
residue field of is easier.)
R.
(The characteristic
The ideal
A" (RG) P
0
has a basis corres-
ponding to the different indecomposables in vertex of
P
case
At of
Actually the projection of an element
A(RG)
onto the linear subspace of
A" (RG) P
corresponding to a given source is determined in [1].
However it is the ideal (or multiplicative)
structure of
A(RG)
which enables the analysis to
go through. However this ideal decomposition is only a particular instance of a much more widely valid (R
arbitrary) and finer decomposition of
flowing out of the coset structure of If
H < G , and if
RH-module, then
IH
G
IH
A(RG)
G.
is the trivial
only depends upon the
conjugacy class of the subgroup
H
in
G
By
the Mackey formula for the tensor product of induced modules we have that
If
H
is conjugate to
Consider the xH .
~
H'
in
-vector space
G B
write on the symbols
We define a multiplication in
B
by
128
FINITE GROUPS
Then
B
is a
-algebra and is called the
~
Burnside algebra.
It is discussed in [2] and [3].
It is semisimple and is the ring direct sum of copies of
~
idempotent
, each ideal generated by an IH
of subgroups ~
x
H
associated to each conjugacy class
H
of
-algebra homomorphism
We have a natural 8 : B
~
~ lH G , with the identity of
the identity of A(RG) 1 :
G.
L
A(RG)
A(RG) B
given by
mapping onto
Thus the identity
1
of
is written as the sum of idempotents 8(I ) H
and we have corresponding ideal
decompositions of
A(RG)
In the particular case when
R
is a complete
local noetherian ring with residue field of characteristic
p,
p'-cyclic extension of a p-subgroup Then the idempotent generator of by the sum of those
8(I H)
G
of
A"p (RG)
where
a complete set of non-conjugate (in of
P
H
G is given
runs through G)
subgroups
which are p'-cyclic extensions of the
p-subgroup
P .
129 REFERENCES 1.
Conlon, S. B., Relative components of representations, to appear in the J. of Algebra.
2.
, Decompositions induced from the Burnside Algebra, to appear in the J. of Algebra.
3.
Solomon, Louis, The Burnside algebra of a finite group, J. of Combinatorial Theory, 2 (1967), 603-615.
CENTRALIZER RINGS AND CHARACTERS OF REPRESENTATIONS OF FINITE GROUPS C. W. Curtis and T. V. Fossum
Throughout this paper finite group, KG
and
KH
H
G
will denote a
an arbitrary subgroup of
G
will be the group algebras of
G
and
H
respectively, over an algebraic number field
K
which is a splitting field for both
KH
KG
and
The first main result (Theorem B) gives some
orthogonality relations in the centralizer ring of a representation of representation of
G H
induced from a linear This result is applied to
give a new arithmetical result (Theorem C) on the degrees of irreducible characters of Let
~
G
1
be an irreducible character of
H.
1. The result on the degrees of characters was obtained independently, and by a different method, by Gordon Keller. 131
132
FINITE GROUPS
Then
W
where
e
is afforded by the left
KH-module
is a primitive idempotent in
centralizer ring
E = eKGe
an algebra with identity simple subalgebra of
~
HO~G
KHe ,
KH.
The
(KGe, KGe)
is
e , which is a semi-
KG
Representations,
characters, and degrees of characters are defined for
E
as well as for
KG
irreducible character of
is the multiplicity of
The character
LEMMA.
character of restriction character
E /;'E
to G
E
/;,
in
ep
wG
of
is an
.
is an irreducible
E
ep
is the
of a unique irreducible
such that
We remark that for
/;,
KG ,
i f and only if
of
/;,
If
( /;,
,
wG )
> 0
.
S-rings, this theorem is
due to Tamaschke [6]. The following notations will be used in the remainder of this paper: x H
=
x
-1
HX,
for
x
E
G ,
133
[H
ind x
We shall also assume that of
is a linear
~
H ; then the primitive idempotent
that
KHe
affords
THEOREM A.
be a linear character of
~
KH-module
= U.lE I
KHe
by (*)
Let
distinct
(H,H)-double cosets
X
E
G
O.l
,~
O.
J
x
=
H
e
x
H
H ,
is given are the
O. l
in
G
Let
such that for some
j
on
~
where
, where the
be the set of indices
I
suc~
can be given explicitly by
~
Let
afforded by the
C
KH
E
\ 'I>(h-l)h. = I Hj-l LhEH '¥
e
J
e
characte~
This condition
depends only on the double cosets and not on their representatives. and let
=
a.
J
For each
j
(ind x.) ex . e J
{a. J
j
E
J}
form a basis for
{D. J
j
E
J}
is invariant under
=
b.
J
(ind x .) ex . J
J
-1
e.
{b.
E
: j
J
J
and
{(ind x.)-lb.} J
respect to the form
J
(x, y)
EO.
J
J
The set
eKGe . 0 ~ 0- 1 E J}
The algebra
Frobenius algebra and, letting {a .}
X.
The elements
J
there is a second basis
pick
E J
xl
E
=
so that
where is a 1 , the bases
are dual bases with
=
Sl
I
where
FINITE GROUPS
134
then the constants of structure algebraic integers in
are all
K.
The computations in the proof are similar to the arguments in Section 1 of [3] and will be omitted.
THEOREM B. ~
Let
be an irreducible character of
(~,
that
Assume the notations of Theorem A.
> 0
1jJG)
G
such
Then
[G ·.H]-l"'(l) S
\' L'
JE
('In d x, )-l"'(b) a.
J
J
L,
•
J
J
where central primitive idempotent in to
~
of
G
If
~
and
~'
both appearing in
KG
corresponding
are irreducible characters 1jJG
with positive
mUltiplicity, then the following orthogonality relations hold:
~
~'
The proof of the orthogonality relations follows from the observations that
135 I:;(E(I:;)e) I:;
t-
I
I:;
( 1:;,
1jJ
G
and
)
I:;
I
o
(E (I:;) e)
. In case
VG ) =
(1:;,
1 , the element
Theorem B is a primitive idempotent in that
KGE(I:;)e
THEOREM C. Let
I:;
that
if
affords
I:;
(by Janusz
KG
in
such
[5]).
Assume the notations of Theorem A.
be an irreducible character of i;;(e)
E(I:;)e
=
I:; (1)
\jJ
(1:;,
G
)
> 0
[G:H]~cm.
JE
J
C(l)
(ind x,) J of
{ind x,} J
such
Then
.
\ [G:H]c(e){L' J JE
Moreover the degree
G
C
-1
I:;(b·)c(a.)} J J
-1
divides
.
The first statement follows from Theorem B. For the second statement, suppose valuation on
K
p
rational prime, and let
R
in
v
corresponding to
principal ideal domain.
=
is a
extending the p-adic valuation On
the rational field, where
K
v
is an arbitrary be the valuation ring Then
R
is a
By Theorem A,
Ra, is an R-order in E containing the J set {b. : j E J} is One can show that I:; (b . ) J J in R for each j E J Let
E'
LjEJ
136
FINITE GROUPS 9, cm .
{ind x.}
]EJ
. L]EJ
w == Then
w
E
But then
p
Q
E'
R
9, I;, (b . ) a . ]]
, and by Theorem B,
It follows that
, since
0:
-1
]
E
for
where
(ind x.)
is in
central in E
and write
]
0.
I;,
w
(w) ==
is ( I;, ,
wG)o:
is a splitting field for
K
== I;, (1) -l(G:H] 9,
is in
is the rational field.
was arbitrary we conclude that
Q
n R
Since the prime 1;,(1)
divides
(G:H]9, Remarks.
Ito's
2
Theorem follows as a corollary
to the above result.
Moreover both extreme cases
in the divisibility formula can occur, in the sense that if
G w
is irreducible,
\jJG(l) == (G:H]
while in the case of the Steinberg character a Chevalley group
]E
[G:H]
of is
[1] ,
relatively prime to X(l) == 9,cm. J
X
, and
{ind x.} . ]
The theorems were motivated partly by the apparent usefulness of centralizer rings of induced representations in the problem of computing the irreducible characters of the Chevalley groups 2.
See [2],
(53.18), p. 365.
137
(see [4],
[7]).
REFERENCES 1.
Curtis, C. W., The Steinberg character of a finite group with a
(B, N)-pair, J. Algebra
4 (1966), 433-441. 2.
Curtis, C. W. and Reiner, I., Representation Theory of Finite Groups and Associative Algebras, Wiley (Interscience), New York 1962.
3.
Dade, E. C., On Brauer's second main theorem, J. Algebra 2 (1965), 299-311.
4.
Gelfand, I. M and Graev, M. I., Construction of irreducible representations of simple algebraic groups over a finite field, Doklady, 147,
(1962),529-532; Soviet Math.
3 (1962). 5.
Janusz, G., Primitive idempotents in group a1gebras, Proc. Amer. Math. Soc. 17 (1966), 520-523.
6.
Tamaschke, 0., S-rings and the irreducible representations of finite groups, J. Algebra
138
FINITE GROUPS 1 (1964),
7.
215-232.
Yokonuma, T., Sur le commutant d'une representation d'un groupe de Chevnlley fini, C. R. Acad. Sci. paris,
264 (1967), 433-436.
University of Oregon
SOME PROPERTIES OF THE GREEN CORRESPONDENCE WaIter Feit
Let
p
>
0
be a prime.
Let
R
be a
complete local domain whose maximal ideal principal such that Let
K
R/(n)
( n)
has characteristic Thus
be the quotient field of
R
either a field of characteristic
p
discrete valuation ring.
be a finite
Let
G
R
is p . is
or a complete
group. By a module we will always mean a finitely generated right module. If v
V
is an
R[G]
module and
denotes the image of
Similarly R[G]
=
-
R
= R/(n)
R[G]/(n)R[G]
v
in
V=
v
E
V
then
V/(n)V .
and ~
R[G]
We will freely use standard terminology and notation. 139
140
FINITE GROUPS Let
G.
If
S)
be
V, Ware
if there exist V
V'
~
W
z
V' m W' V
is
o
a nonempty set of subgroups of R[G]
R[G]
m W'
modules write
modules
V', W'
such that
and for every component H = H(V ) O
there exists
R[H]-projective.
S)
E
= W(S))
V
V
of
o
such that
Since the unique
decomposition property (Krull-Schmidt theorem) holds for
R[G]
modules [3,
(76.26)]
well defined equivalence relation. V
is said to be
some
x E G
If
V
I f Ax
R[~]-projective.
write
this is a
=
O(~)
E
S)
A E S) . G
Let
P
be a p-subgroup of
be a subgroup of
G
with
G
mrG(p)
C
the following sets of subgroups of
l=
l(P, H) = {AlA ~ P
x
E
~=
E
H
Define
H G.
n pX
for some
n pX
for some
G - H}
2)= 2)(P, H) = {AlA
x
and let
~
H
G - H}
~(P, H)
=
{AIA~ P
and
A
t
l}
G
Green [5] has proved the following basic resul t.
THEOREM.
There exists a function
f
from the
for
141 set of all isomorphism classes of R[G]
R[P]-projective
modules into the set of all isomorphism
classes of
R[p]-projective
R[H]
modules which
has the following properties. (i) (ii) ( iii)
f(V
(iv) (v)
Let
= f (V1)
®
V2)
-
VH (ID)
V
f(V)
2
V - o (l)
be an indecomposable ~ ;
then
a common source and vertex. R[H]
f (V )
f (V1) ® f(V 2 ) (I) f(V)G and
-
module with vertex in
posable
Gl
i f and only if
(0 )
f(V
l f (V)
V ) 2
$
l f (V)
If
V
vlw
W if and only if
In view of (v) the mapping
V (l) R[G] f (V)
have
W is an indecom-
module with vertex in G
and
-
~
then
or equivalently
f
yields a one
to one correspondence between the set of isomorphism classes of indecomposable ~
vertex in
f
with respect to If
V
moduies with
and the set of all isomorphism classes
of indecomposable We will call
R[G]
R[H]
or this the Green Correspondence (G, P, H)
is an
the submodule of
~
modules with vertex in
R[G] V
module let
InvG(V)
consisting of invariant
be
FINITE GROUPS
142
elements.
H
If
is a subgroup of
G
define
Inv (V) -+ InvG(V) by NG,H(v) I vx.l H {x. } is a (right) cross section of H
NG,H where
in
l
is the relative
G .
norm and is
(G, H)
independent of the choice of cross section. For a nonempty set and an
R[G]
Clearly
module
HO(G,~, V)
V
H
of subgroups of
G
define
is an
R
module.
In this notation D. G. Higman's criterion for relative projective modules can be formulated as follows.
THEOREM.
Let
V
be a subgroup of if and only if
be an
G
R[G]
Then
V
module and let is
H
R[H]-projective
HO(G, {H}, HomR(V, V))
=
(0)
The following result can be proved by making use of Higman's theorem. THEOREM l. let
H
Let
X, V,21
Let
P
be a p-subgroup of
be a subgroup of
G
with
be defined by (*) •
OOG(P) Let
Green correspondence with respect to
f
G C;;;;
and
H be the
(G, P, H)
143 Let
V, VI' V
R[G]
be indecomposable
2
modules.
Then
°
for any automorphism
(i)
of
R[P]-projective
R[G]
with
=
RO
R
and
GO
=
G
where
fO
is
the Green correspondence with respect to (G, pO, HO) (ii) where
If
V*
V
is R-free then
is the contragradient of
( iii)
f(Hom
(iv)
HO (G,
( v)
H (G,
R
~H
If
in a block
V
B
HO(H,
V,
Hom
(ii) that
V
HO(H,
V,
module.
~
Hom
Hom R (VI' V )) 2
H
V. (f(V ) ,f(V )) (x). l 2
V,
Hom
R
then
~
(f(V), f(V)))
BG
~
and
f(V)
is
is defined and
V
has the same meaning as in [2]).
(i) R[G] R If
K
R
f (V) *
~ HO (H, iJ), f (V))
V)
°(H,
of
COROLLARY 2. irreducible
:: Hom
has vertex in
-G B .
is in
2
x, x,
°
(vi)
(VI' V ))
=
f (V*)
If
V
module with vertex in
(f(V), f(V))) V
is an absolutely ~
=R
is an R-free
R[G]
module such
is absolutely irreducible then R
(f(V), f(V)))
then
is a cyclic
R
FINITE GROUPS
144
COROLLARY 3.
Let
p
a
Ip :AI
min
Ad that
V
is an R-free ~
and
If
dim
has vertex in irreducible.
R[G]
module such that
V 0 K K
Suppose V
is absolutely
(V 0 K)
<
1P?
then
f(V) ~ K
is absolutely irreducible. The difficulty with applying the results of Theorem 1 and Corollary 2 is due to the fact that generally there are many
R[H]
modules which
satisfy the conclusions of the Corollary.
This is
illustrated for instance by the following result which follows easily from Schanuel's lemma.
LEMMA 4. R-free
For R[G]
i
=
1, 2
let
modules with
Vi' Wi , Ui
be
U.
projective such
~
be a nonempty set
l
that V. l
is an exact sequence. of subgroups of
G
i
-+
0
Let then
From now On assume that the following conditions are satisfied.
145
B P
G BG
P
WG(P)
B
and
with cyclic defect group
= B
For any module f
p
in
P
G
is the unique block of V
in
B
V
let
with f(V)
is the Green correspondence with respect
(G, P, G) char K = 0
for
G
is the subgroup of order
where to
is a block of
G
K
and
R
are splitting fields
and all its sUbgroups.
In [4] Dade has proved several basic results concerning the characters of
G
which are in
These results generalize those of [1] [8].
B .
His
procedure is inductive and in particular in case G =
G
he gives a complete description of all
R[G]
indecomposable
B.
modules in
By making use
of these results and Theorem 1 it is possible to recapture his results in the general case.
The
argument is based on the following two lemmas.
I
am indebted to D. Passman for a proof of the first.
For
LEMMA 4.
posable length
R[G] n. l
i
=
1, 2
let
V. l
be an indecom-
module whose composition series has If
then
146
FINITE GROUPS
LEMMA 5.
V
If
is an
o
H (G, <1>, Hom R (V, v))
R[GJ 0
module in
~
H (G, <1>, Hom
~
R
B
then
~
-
(V, V))
By extending these arguments one also gets short proofs of the following results.
THEOREM 6. In
B
Let
V
be an
R-free
R[GJ
module
which satisfies the following.
e
Let
(i)
e
Then
V 0 K
be the character afforded by is either irreducible or a
mUltiplicity free sum of exceptional character. (ii) V
The socle of
V
is irreducible.
Then
is uniserial.
THEOREM 7.
(Janusz (6J)
indecomposable modules in
The number of
( i)
B
is
elpj
where
e
is defined as in (4 J . (ii)
in
B
is multiplicity free.
( iii)
R[GJ where
The socle of an indecomposable module
Let
U
module in Ll
B L
and
2
be a projective indecomposable Then
Rad(U)
=
are uniserial and
L
+ L
l
L
l
2
n L2
is
irreducible.
If furthermore there exists an R-free
R(GJ
V
module
such that
V
is the irreducible
147 module corresponding to
U
then
U
is uniserial.
The following result was first proved by the
Ipl =
author in case
p
and then by Rothschild [7J
in general.
Let
THEOREM 8.
module in
B
some integer
V
Then a
with
be an irreducible dim V
lal
~
=
a
(mod
R[GJ
Ip!)
for
e .
In particular Theorem 8 implies that every irreducible
R[GJ
module in
Then Theorem 8 of [4J Theorem 2 (i)
B
has vertex
P .
is a simple consequence of
applied to the Green correspondence
with respect to
(G, P, WG(P))
REFERENCES
1.
R. Brauer, Investigations on group characters, Ann. of Math. 42
2.
(1941),
936-958.
, Zur Darstellungstheorie der Gruppen endlicher Ordnung, 11. 25-46.
Math Z. 72
(1959)
148 3.
FINITE GROUPS C. W. Curtis and I. Reiner, Representation theory of finite groups and associative a1gebras, Interscience, New York (1962).
4.
E. Dade, Blocks with cyclic defect groups, Ann. of Math. 84 (1966), 20-48.
5.
J. A. Green, A transfer theorem for modular representations, J. of Algebra 1 (1964), 73-84.
6.
G. J. Janusz, Indecomposable modules for finite groups, to appear.
7.
B. Rothschild, Degrees of irreducible modular characters of blocks with cyclic defect groups, Bull. A. M. S.
8.
(1967), 102-104.
J. G. Thompson, Vertices and Sources, J. of Algebra 6 (1967), 1-6.
Yale University
INDECOMPOSABLE MODULES FOR FINITE GROUPS
G. J. Janusz
§l.
INTRODUCTION
We first describe a construction for a class of finite dimensional algebras over a field K.
It turns out that the algebras are symmetric
algebras with only a finite number of indecomposable modules.
Moreover all the indecomposable modules
can be described quite explicitly in terms of a graph from which the algebra is originally constructed. The application of this work is to the construction of the indecomposable modules for a finite group p.
G
over a field
K
of characteristic
The two sided ideal direct summand
group algebra
K(G)
Al
which corresponds to a 149
of the
150
FINITE GROUPS
p-block having cyclic defect group is shown to be an algebra of the type constructed above. all the indecomposable
Hence
K(G)-modules in this block
can be described in terms of the graph associated with the block.
§2.
l
CONSTRUCTION OF A CLASS OF ALGEBRAS We start with a graph
edges
E, E',
such that that is
j/ j/
and
j/
e + 1
consisting of vertices
e
P, Q,
is connected and contains no cycles7
is a tree.
One vertex is selected and
called exceptiona17 the others are called non-exceptional.
The exceptional vertex is
assigned a multiplicity
m > 0
there is a fixed imbedding of
We also assume j/
into the plane.
This imposes a cyclic ordering of the edges which contain a given vertex
P
distinct edges containing the vertex
P.
El' E 2 , E 3
El' E 2 , E 3
are in proper P-order if
We say
is the clockwise order of the three edges around P.
If the order
El' E 2 , E 3
is counterclockwise
1. Proofs of the results stated here will appear in Annals of Mathematics.
151 we say the edges are in improper P-order.
57 is used to construct an algebra
The graph A
over a field
each edge
E
K
Y
of
in the following way.
,
eE
For
is an idempotent in
A
and
L e Esy E
1
is an orthogonal decomposition of the identity of A.
We define integers
CEE ,
(to become the
Cartan numbers) by is empty
0
if
E nE'
= 1
if
E n E' = P
CEE '
i exceptional
vertex = m
if
E n E' = P
exc
=
( *)
exceptional vertex CEE
2
if
m + 1
P if
E exc ~
P
E exc s
.
The full multiplication table for
A
is
determined in the following. (M)
and
A
has a K-basis consisting of elements subject to these conditions:
x
E
i
152
FINITE GROUPS (i)
For each edge XE
element
in
~ c-l ~ 0
(ii) with
E
~
element
XE
A c
E
such that 0
5
of
where
eEAe E
=
c
YEE'
but
E
in
A
n E'
=
and
K[X E ]
CEE
For each pair of edges E'
there is an
E, E'
Y
of
non-empty, there is an
such that
YEE,X E ,
n E' = {Q}
E if E
E
n
(v)
=
QP,
P
P exc
~
T
E' = {Q}.
if
El' E 2 , E 3
al?e
distinct and in proper P-order for some vertex
xEl YE
P
if E E E l' 2' 3 1 E3 are distinct and in improper P-order for some vertex
P
have a common vertex.
153 Now let
=
UE
Ae E
and
FE
=
uE/rad(A)U
E
denote the projective indecomposable module and the irreducible module respectively corresponding E
to the edge
Y.
of
multiplicity of
FE
Then
C , EE
of ( *) is the
as a composition factor of
UE ' If
E
=
PQ
is an edge of
contains two submodules
VE,P
Y
then
U E
and
VE,Q
which
are uniserial (submodules are linearly ordered by inclusion) and such that rad(A)U
VE,P + VE,Q of VE,P
then
FE
I
FE'
minimal submodule of is a composition factor of
is a composition factor of P
and only if VE,P' VE,Q
U E
n VE,Q
Moreover if
maximal submodule
E
is a vertex of
E'
VE,P
U . E U ' E
if
Since
are uniserial, their composition
factors appear in a unique order (there is a unique composition series).
This order is
precisely the P-order of the edges of
Y.
We shall use a special notation for certain submodules of
VE,P
edge such that
FE'
or
Let
El
be an
is a composition factor of
154
FINITE Let
=
M(E', E, n)
uniserial submodule of
DE
M denote the
such that
at the top of a composition series for FE'
appears
n
times in the series.
necessarily unless vertex. E
If
E'
contains
P
n
E'
n E
E
M
Notice
n
=
When
M(E, E, n)
to be
the (unique) uniserial module with
FE
~
1
as top and
bottom composition factor and multiplicity
require
and
m
thenc1 · ~ n Pe x
the composition series.
appears
FE'
contains the exceptional
we can define
exc
GROUPS
n
in
In this case we shall
n';' 2 .
We can now describe all the indecomposable A-modules. EO,El, ... ,E
Consider chains of edges k
,
k
>
1
of one of the following
types.
0.-.0----0
C;T)
1
O_O\'E
h
' \ Eh + l yO~o
(§)
2
I'E o_o~o
0_0
h+2t h+2t+l
Eh + t
... o_op exc Eh + t + l
155 In
the branch point
(§)
2
P
we allow
o=
Q
cannot be
Q
or
Let
D
P
but
exc
be the
set of even integers or odd integers in
M.
M (E
1
M.
Let
{O/l/ ••• /k}
1 In. )
i
E
1 lE. In. )
i
~ D
E.
'I
1
M (E.
1-
be the uniserial module
1
1-
1
1
cp.
Select homomorphisms irreducible module
1
1
FE.
E
mapping i
E
F or
D
~
i
or
D
Ei - l
~
i
F
be a monomorphism
Let
W·1
or
FE.
into
Ei - l
M.
.
In the direct sum
define the submodules
X, Y
cp.
X
1
FE.
according as
according as
1
1
D
onto the
1
or
.
D
M.
mapping
1
i
D
i
E
D
M Ell l
...
Ell M k
by (m. ) 1
and
0 < i
< k}
1
Y
1jJ. (f) 1
f
o Since all the
THEOREM.
<
E F _ i l
liE D
I
i} M i
are reducible it follows
The quotient module
W
X/Y
is
Y
~
X .
FINITE GROUPS
156
indecomposable and every non-projective, reducible, indecomposable A-module is isomorphic to a module of this form. modules is
The number of indecomposable
e(em + 1) .
APPLICATION TO GROUP ALGEBRAS
§3.
Let
G
be a finite group and
with cyclic defect group.
Bap-block
The recent results of
Dade (2J and the earlier results of Brauer (lJ show that the relations between the complex irreducible characters and the modular irreducible characters in
B
which is a tree. vertex of
can be described by a graph (See [lJ for details.)
p
and let
K(G)
One
31 corresponds to the family of
exceptional characters, Let
jY
K Al
m
in number, in
B.
be a splitting field of characteristic denote the ideal direct summand of
which corresponds to
B.
Let
sl"",se
be a set of orthogonal idempotents in
Al
such
that every projective indecomposable AI-module is isomorphic to one and only One of Then constructed from the graph
A = sAls
As.l
Set
is an algebra
31 (subject to a
157 suitable imbedding in the plane) as in the previous section. then
AlE
If ~
W is any indecomposable A-module, W is an indecomposable
A
K(G)
) mOdule.
A l
(hence
This correspondence preserves the
lattice of submodules, etc.
Moreover every
indecomposable
M
K(G)-module
has this form by taking
in the block
B
W = EM
REFERENCES 1.
R. Brauer, Investigations on group characters, Ann. of Math. 42 (1941), 936-958.
2.
E. Dade, Blocks with cyclic defect groups, Ann. of Math. 84 (1966), 20-48.
RELATIVE GROTHENDIECK GROUPS T. Y. Lam and I. Reiner
1.
Representation theory has played an important
role in many problems about finite groups.
While
character theory is adequate for many applications, it would obviously be better to be able to deal the representations themselves.
wit~
The use of relative
Grothendieck groups is an intermediate procedure, giving more information than character theory, but bypassing some of the difficulties in the study of the actual representations. Let where
p
be a field of characteristic
~
~
0 , and let
G
p,
be some finite group.
By "G-module" we mean a left finitely generated ~G-module.
be a full set of s non-isomorphic principal indecomposable G-modules, and
Let
Fl, ... ,F s
Ul' ..
"U
the irreducible G-modules, where 163
164
FINITE GROUPS is the unique minimal submodule of
i'.
~
Now let and
o
7
S L
C
u.
~
be some category of G-modules,
some collection of short exact sequences 7
M
7
N
7
from
0
Form the free
C
abelian group on the symbols
[t-1]
, where
ranges over the isomorphism classes in
M
C , and
factor out the subgroup generated by all expressions [M]
-
[L]
-
[N]
coming from sequences in
S.
The
factor group thus obtained is called a "relative Grothendieck group." Let
H
be some subgroup of
G, and take
to be the category of all G-modules, and
S
C
the
set of H-split exact sequences of G-modules. Denote the resulting Grothendieck group by a(G, H)
For example,
a(G, 1)
s
L
ffi
Z[F.] ~
i=l
~
ring of generalized Brauer characters.
At the other extreme,
a(G, G)
is the "Green ring"
consisting of G-modules relative to direct sums. (Note that forming tensor products of modules makes a(G, H)
into a ring.)
Secondly, take
C
to be the category of
(G, H)-projective G-modules
(these are direct
165
xG
summands of induced modules and take
S
k(G,
H)
k(G,
X
=
H-module),
to be the set of all G-split exact
sequences from by
,
C.
Denote the Grothendieck group
in this case.
Thus
s Ell
L
1)
k(G,
Z [D.]
i=l
G)
= a(G, G)
l
The Krull-Schmidt theorem for G-modules implies that both
a(G, G)
and
k(G, H)
are
Z-free.
The Jordan-Holder theorem tells us that
a(G,
is Z-free.
that M
1)
a(G, H)
It is not obvious, however,
is Z-free in general.
A G-module
is called "H-simple" if no G-submodule of
is an H-direct summand of a(G, H)
La.l
M
M
Every element in
is expressible as a linear combination M.
[M. ]
l
l
H-simple.
Such
expressions are in general not unique, so the H-simple G-modules need not form a Z-basis for a(G, H)
One of our main results is the assertion
that under suitable hypotheses,
2.
There is an additive map
given by
K[M]
homomorphism."
=
[M]
When
a(G, H)
K:
k(G,
is Z-free.
H)
+
a(G, H)
, called the "Cartan
H = 1 , this is the usual
FINITE GROUPS
166 Cartan map K
where
\' c .. [F.]
[U . ]
lJ
L
l
=
(c .. ) lJ
J
Cartan matrix of
G.
An argument
due to Con1on shows
THEOREM 1. K:
If
k(G, H)
H~
a(G, H)
-+
G , the Cartan map is monic, and its cokernel
is a p-torsion abelian group. Since
a(G, H)
is unchanged when
H
is
replaced by its Sylow p-subgroup, we may assume H
is a p-group.
Hereafter, we make the following
restrictive hypothesis:
H
~
element
G,
x
H
of order
nJ
=
{u
E U.
J
1 < j
THEOREM 2. 1 < j
< s}
=
r:okernel of
h == P
w = x-I
Define
v .
cyclic p-group generated by an
n-l
and set
~H,
E
w
e
u
E
F.} J
<
h ,
< s}.
The set
{[V
.]
:
1 < n =< h
nJ is a Z-free basis for K:
1 ;;, n
k(G, H)
to the direct sum of
h
-+
a(G, H)
a(G, H)
, The
is isomorphic
copies of the cokernel of
167
the ordinary Cartan homomorphism k(G/H, 1)
+
a(G/H, 1)
Sketch of proof.
M
Let
be a G-module, and let
r-l w M 'I 0 For F any irreducible k r-l submodule of w M , define M* by the pushout
wrM
=
0
,
diagram
Since the ring
rcH/wrrcH
is self-injective, both
sequences
o
+
M
+
are H-split.
M*
+
M*/M
+
0 ,
This yields a relation in
a(G, H)
The result then follows by an induction argument.
COROLLARY. for each
Let r,
M, N
be G-modules such that
o ~ r ~ h - 1,
have the same Brauer character.
wrM
and
wrN
Then for each
FINITE GROUPS
168 G-module and
T
and each
wr(T 0 N)
r , the modules
wr(T 0 M)
have the same Brauer character.
Under the additional hypothesis that
3.
a semidirect product
G
H'A, we may view each
is F. J
as an irreducible A-module, and form the induced module
(F.)
G
J
.
Let us set 1
THEOREM 3.
The
Z-basis for
hs
~
n
symbols
a(G, H)
<
h ,
[M
.]
nJ
l<j<s.
form a free
M. is H-simple. nJ The matrix of the Cartan homomorphism K:
k(G, H)
-+
a(G, H)
as a direct sum of of
Each
can be expressed naturally h
copies of the Cartan matrix
A.
COROLLARY.
If
G = H
x
A
(direct product) ,
there is a ring isomorphism a(G, H)
4.
'" a(A, 1) 0
z
a(H, H)
Reverting to the hypotheses of section 2,
assume now that
G
contains a p-complement
There is a commutative diagram
B .
169
8
a(G, H)
>
a(HB, HB)
1
1
a(G, 1)
with the
if
8
)
a(B, 1)
0
given by restriction.
8's
THEOREM 4. epic.
8
is epic if and only if
8
Further,
8
H)
~
Let
a(H, H)
COROLLARY 2.
M
is
is an isomorphism if and only
G
be any p-group, and
any normal cyclic subgroup of
Let
0
is an isomorphism.
0
COROLLARY 1.
~(G,
8
M, N and
N
be
G.
H
Then
by restriction.
Assume
8
0
is an isomorphism.
(G, H)-projective G-modules.
Then
are G-isomorphic if and only if they
are HB-isomorphic.
5.
Under the hypotheses of section 2, we may
describe the ring structure of s $
R
L
i=l
Z[F
i
]
C
a(G, H)
a(G,H)
thus.
Set
170
FINITE GROUPS
a subring of
l-representation of
v
[V
k
k
and let
G
F
l
be the
Define
]
Hp ,1 V nj
where the a(G, and each
,
a(G, H)
are as in section 2.
Then
H)
v
k
satisfies a
P
th
degree equation
Furthermore,
a(G,
contains no nonzero nilpotent elements.
H)
ISOMETRIES AND CHARACTERS OF FINITE GROUPS
w.
F. Reynolds*
It is desirable to be able to obtain information about the characters of a finite group G
from information about the characters of a
subgroup
H ; in particular, if
G
is simple,
such information is often useful in characterizing G
(see [15],
papers).
[7],
[1],
[9], and many other
Frobenius used this approach in proving
the basic theorem about Frobenius groups [9, p. 140, Theorem 5.1],
[5, p. 104,
(20.2)].
Brauer and
Suzuki developed this approach into the study of trivial intersection subsets (T. I. sets) [6, §23],
[9, p. 134].
[16],
Such sets can be described
* This work has been supported in part by the National Science Foundation under Grants GP-4317 and GP-6954. 171
FINITE GROUPS
172 as subsets
A
of
H
such that
(1)
A
(2)
any two elements of
conjugate in
G
A
which are
are conjugate in
for every
(3)
H,
is a union of conjugate classes of
a
CG(a)
A,
E.:
H , ~ H
•
Under these assumptions, the induction ~-space
mapping gives an isometry of the of class functions On
1fG
into the space
G
G (a , 6 )G =
i. e.
H
~~(A)
which vanish outside
of class functions on
6)H
(a,
for all
a,
6
A
G, E.:
~~(A)
furthermore if ao (a) a
(4 )
G
G,
E.:
X
a
and {
a
are conjugate in
if
x
E.:
G
is not conjugate to
any element of
a
G
a
A.
is a generalized character (i.e. a
~-cornbination
whenever
A , and if
=
(x)
Of course
E.:
x
of irreducible characters) of
is a generalized character of
G H.
To make use of this result, it is usually necessary to have a set of irreducible characters of
H
with a property called coherence, which we
shall not define here; see [7], [6,
§31],
[9, p.
148]
[12],
[16],
Then there is determined
G,
173 a corresponding set of irreducible characters of G
about which considerable information can often
be obtained.
These characters of
G
are usually
called exceptional characters; this strange name originated in the situation of cyclic defect groups (see [5]
for recent results), where they
correspond to an exceptional vertex of the associated tree.
The study of exceptional
characters has been especially fruitful in characterization theorems, and related methods lead to theorems about existence of normal subgroups [2, Theorem 1],
[12, Theorem 2.3].
However, we shall restrict our attention here to the existence of isometries rather than to the uses of them. In the odd-order paper [7, §9], Feit and Thompson found it necessary to study tamely imbedded subsets
A
of
H , which satisfy a quite
complicated set of assumptions weaker than (1), (2),
(3).
They obtained results analogous to
those in the trivial intersection case, but the isometry is no longer induction.
In their approach
it is clear that generalized characters are mapped on generalized characters, while it takes a fair
FINITE GROUPS
174
amount of work to establish that the mapping is an isometry. In [4]
(cL
[6,
§33]), Dade generalized and
at the same time simplified the Feit-Thompson isometry.
In Dade's situation, a set
is given, along with slightly as in [13] (1),
(2),
A.
and
of primes
Generalizing
(cf. also [12]), we can replace
by the following assumptions:
(3)
(1')
G, H,
IT
A
is a union of IT-sections of
H
(where two elements are said to be in the same c-section if their IT-parts are conjugate), (2')
any two IT-elements of
conjugate in
G
has a normal IT-complement
I(a)
which are
H,
are conjugate in
for every IT-element
(3')
A
a EA,
C
G
(a)
, and
Under these assumptions, there exists an isometry ~G-
a
,IT , where
functions on
of the space :~H
H
,IT
_'GH (A) n
~H, IT
into
is the space of complex-valued
which are constant on each
IT-section, given by
175
if
a (a) x (4 ' )
a
o
x E G and
a
,
0
G
0
if
,
if the IT-section of is disjoint from
Here
, and
are in the same
IT-section of
(x)
a E A
in
x A
G
.
carries generalized characters to general-
ized characters, and satisfies the following weaker analog of Frobenius reciprocity: (5 )
a (a , e) G If
IT
= (a, e/H ) H
'
a
E
~·H ,IT
'
e
E
~G ,IT
contains all the prime divisors of
G , this theorem reduces to that of Brauer and Suzuki.
The hard part of the proof is showing
that
carries generalized characters to
0
generalized characters; this is proved by a very ingenious and (to me at least) rather mysterious Mobius-type formula [4,
(3.5)]
All of this theorem except the statement about generalized characters (which is crucial in applications) still holds if (2') is replaced by the weaker assumption (2")
for every IT-element
a EA,
176
FINITE GROUPS
It is therefore natural to conjecture that the entire theorem holds in this case.
It would be
good to know this since factorizations similar to that in (2") occur, for example in a theorem of Glauberman [9, p. 279, Theorem 2.11]. At this point the theory of modular representations comes into play.
Some connections
between block theory and isometries had been found by Gorenstein and Walter [10, proposition 25J and by Wong [17], but these results were not aimed at proving results like the above theorem.
In his
method of columns [1], Brauer used results about the principal p-block
BO(G)
of
G
to obtain
results analogous to those of exceptional character theory.
The connection between Brauer's
and Dade's situations was made explicit in a special case by Glauberman [8,
§ 3],
and has been
exploited by Feit (unpublished, but see [6, pp. 174-175J).
Using these ideas, Alperin and
Gorenstein (unpublished) proved the case
TI
=
{p}
of the conjecture, and later an independent proof was given in [14J :, E.Jf (A)
H
follows:
n :r;H, p ' if
aG
The idea here is that for any aO
can be described as
is expressed as a linear
177 combination of the irreducible characters of then
ex
o
G ,
is the subsum containing precisely those
terms involving irreducible characters in
BO(G)
Note that the generalized character property follows at once from this description.
The proof
depends on Brauer's results On principal blocks [1, I, Theorem 3], and is quite different from the earlier proofs. that
ex
E
.cc,·H·
,p
Furthermore the requirement
can be replaced by the weaker
requirement that the expression of the irreducible characters of
H
ex
in terms of
should involve
(Here (4') must be
only characters In
Ba(H)
changed slightly.)
This weakening also facilitates
the treatment of coherence.
It seems likely that
this requirement can be weakened still further so as to replace
Ba(H)
and
Ba (G)
by certain
unions of p-blocks, making use of [3, §6] ,and some of Feit's unpublished work. It remains to be seen whether the conjecture, or something like it, holds for general
TI
Per-
haps Iizuka's theory of TI-blocks [11] will be of use here.
178
FINITE GROUPS
REFERENCES 1.
R. Brauer, Some applications of the theory of blocks of characters of finite groups, J. Algebra, I, 1 (1964), 152-167; Il, 1 (1964), 307-334; I l l , 3 (1966), 225-255. , On quotient groups of finite
2.
groups, Math. Z. 83 (1964), 72-84. , On blocks and sections in finite
3.
groups, I, Amer. J. Math. 89 (1967), 1115-1136. 4.
E. C. Dade, Lifting group characters, Ann. of Ma th.
5.
7 9 (196 4), 5 9 0- 5 96 . , Blocks with cyclic defect groups,
Ann. of Math. 84 (1966), 20-48. 6.
W. Feit, Characters of finite groups, W. A. Benjamin, Inc., New York, 1967.
7.
W. Feit and J. G. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775-1029.
179 8.
G. Glauberman, A characterization of the Suzuki groups, Illinois J. Math. 12 (1968), 76-98.
9.
D. Gorenstein, Finite groups, Harper and Row, New York, 1968.
10.
D. Gorenstein and J. H. Walter, On finite groups with dihedral Sylow 2-subgroups, Illinois J. Math. 6 (1952), 553-593.
11.
K. Iizuka and T. Nakayama, A remark on orthogonality relations in finite groups, Nagoya Math. J. 20 (1962), 185-194.
12.
H. S. Leonard, Jr. and K. K. McKelvey, On lifting characters in finite groups, J. Algebra 7 (1967), 168-191.
13.
W. F. Reynolds, Sections, isometries, and generalized group characters, J. Algebra 7 (1967), 394-405.
14.
, Isometries and principal blocks of group characters, to appear.
180 15.
FINITE GROUPS M. Suzuki, The nonexistence of a certain type of simple groups of odd order, Proc. Amer. Math. Soc. 8 (1957), 686-695.
16.
J. H. Walter, Character theory of finite groups with trivial intersection subsets, Nagoya Math. J. 27
(1966), 515-524;
correction, same J. 30 (1967), 309. 17.
W. J. Wong, Exceptional character theory and the theory of blocks, Math. Z. 91 (1966), 363-379.
Tufts University
ON THE INTEGRAL REPRESENTATIONS OF AN ORDER Hans Zassenhaus
Let 1t field
F.
be a dedekind ring with quotient Let
H
be an absolutely semi-simple
hypercomplex system over H
-0 and its
of
H
called an of
-t'-submodule of
H
~-rank is equal to the F-dimension
~-order
crof
~-submodule
The dedekind
H
is
if it contains the unit element
and if it is closed under multiplication.
For example the group ring G
An
is called dedekindean if it is finitely generated
over n
F
of a finite group
~(G]
over the rational integer ring
of the group algebra
of
~(G]
A unital cr:module
M
~ -representation module of generated torsion free
G
181
is a
over
~-order
~
is called an
er
if it is a finitely
~-module.
~-representation modules
~
Two of
are
FINITE GROUPS
182
called (i:equivalent if there is an er-isomorphism of one onto the other one. In this way the "" -representation modules of
(JI
are distributed in arithmetical equivalence
classes.
We try to characterize them by local and
global invariants. ~-rank
One invariant is the
r~(M)
The
algebraic invariants are the invariants referring to the representation theory of To every
Hover
lr-representation module
F M
there
corresponds the proper F-representation space of
H.
Denoting by
r l ,r , ... ,r p 2
FM
a representative
set of the proper irreducible F-representations of H
the F-representation
r
of
H
with
FM
as
representation space is equivalent to a representation p
Lj=l ~jrj integers in
with uniquely determined non-negative
~. = ~. (M/I) , the mul tiplici ty of J
J
r.
J
r As ., -representation space of
minimal non-zero left ideal chosen.
~
0
of
some
H
may be
is a minimal
It follows that
two-sided ideal
r1
H
and that
183
Moreover
H.
contains a set of
J
(i,k
=
1,2, ... ,f ) j
f.
2
matrix units
J
such that
eo"
Jll
D.
I
{x
J
all
H.
E:
X
J
&
for
e.Jl'kx
xe.Jl'k
i,k}
is a division algebra in
H.
J
a ring of matrices of degree
F,
over f.
J
over
=
H.
J
D. J
,
D. J
f .x f , J J
the
elements
e ...
E.
Jll
J
(j
1,2, ... ,p)
are the primitive idempotents of the center of
H
of
D. J
,
is the unit element of
E. J
H,
J
}(H)
as well as
,
is representation space of an irreducible representation
~.
degree
f .m.
of
J J
J
2
H.
J
over its center
,where
of
FINITE GROUPS
184
m. 2 J
~.
the restriction of
to
J
is equivalent to
D.
J
f. -times the regular representation
of
p. J
J
det p. (x)
N. (x) J
(x E
J
is the regular norm of det ~. (x)
Dj
N. (x)
J
J
over
D.)
J
~(Hj)
f· J
the elements
det
subgroup
of the multiplicative group of
1);.
J
the elements
of
J
~ (H j )
over
(l
D.
< p)
< j
=
=
)(H)
the
~.(x)
form a
(O;;iXED.)
J
J
...
t: l + t: 2 +
+
form a subgroup
t: p
t:.
with
a
(H j )
in
J
1jJ.
J
of the unit group
1jJ
isomorphic to the direct product of
1jJ . •
J
Each left ideal
L.
J
is generated over
some 4-representation space example we may set non-zero element of
o-u.
L. =
J
L
j
L.
J
J
of
when
(). u.
J
F
For is some
.
A global invariant of the ;r-representation space
M
of
0- is
exists a submodule
defined as follows:
There
by
185
P
of
FM
]Jj
I I j=l i=l
M
containing
L.8 ..
J Jl when
M
L.
er-monomorphism of
J
The residue class
in
eJl ..
is an
FM.
R,U·1/(j")
of the index
ideal
depends only on
modulo
M
(f
and
course, on the choice of the standard L. ).
~modules
Here we have interpreted the factor module
J
M/M
(and, of
as a module over the center
~ ((J)
of
(J',
a
unital commutative ring and we have formed the order ideal
M :
M
in the sense of E. Steini tz [7].
3(0) We observe that
and that for any
~ -order
0' of
H
containing
0"
we have
when we define the standard
~-representation
space
186
L.
J
of
-
FINITE GROUPS
0-
by means of
=
L. J
Ci j
(l < j
=
In the event that and
(J
Ll =
then
< p)
=
-
(J'
k(M/O')
is an integral domain is an ideal class of
(jrin the ordinary sense which has been defined already by E. Steinitz (7].
THEOREM 1. then two
If the .., -order
(J'"
of
~-representation spaces
is maximal
H
M , M 2 l
(jr
of
are er-equivalent if and only if
~j (M1/~) = ~j
(1)
R(M1/U)
(2)
THEOREM 2. then the
( M2 /(7)
j
<
<
p)
= ~M21U")
If the ,)'-order
(J'
of
f-representation space
indecomposable if and only if F-representation space of
H
M of
FM
is maximal
~is
is an irreducible
H .
An arbitrary "",-order
0-
embedded into a maximal 1) -order (s.
(1
of
-
can be
H
0'"
of
H
(3]).
The conditions (1),
(2) are necessary for
187
():equivalence of the
of~.
M , M 2 l
~-representation spaces
They imply that (l < j
lJ j(tfM 2101
< p)
RJ(TM2 1U;
ffM l
hence
is tT-equivalent to
to establish conditions for the M l
and
M 2
tfM 2
In order
er-equivalence of
it suffices to assume that
(I)
The global problem is reduced under certain conditions to a local problem for the distinct prime ideals 11"2' ...
''1s
~occurring
of
in the
prime ideal factorization
when
)0-
and
v. > 0
denotes the discriminant ideal (s. (1 < i
l
[3])
~ s)
The conditions are (11)
The residue class fields finite
(Ill)
If lJ
j
lJ
j
(i
=
(M1/{fl
(Mll(J) > 1
k
i
= -lJ
1,2, ... ,s)
>
0
or
then either lJ
j
(Mlli)J
= 1
I,
i
are
FINITE GROUPS
188
and
f. > 1 , or
fl.
J
J
(Ml/(J)
f.
=
=
J
m.
1.
J
The condition (11) is always satisfied for ~=
~
~
or if
is the intersection of valuation ~
rings of a finite extension of (x
~/p(x)
or of
an algebraically independent element over the
field
~/p
of
p
elements,
p
a prime).
The condition (Ill) in the case of integral finite group representations for which only one fl. J
is not zero is usually fulfilled, excepting
certain groups determined by Amitsur [1]. using a theorem of Eichler only the case that
H. J
is a positive definite quaternion
algebra over
or that
E. ([)
J
H.
J
is not amenable yet. If several
fl. J
are not zero then, by the
application of the theory of connecting systems ("Verbindungs systeme," s.
[8]) the problem can be
reduced to the case that only one We call
M , M l 2
1
J
is not zero.
properly er-equivalent if there
is a linear transformation determinant
fl.
8
of
FM l
of
which upon restriction to
yields an er-isomOrphism on
M . 2
Ml
This equivalence
relation is a stronger equivalence concept than
189
just er-equivalence. In order to establish the connection between proper er-equivalence and er-equivalence one determines the group
of all CI-automorphisms of
M l
as the unit group of the ring
which is the same Horn ((M , M ) I ()) l l
formed by the cr-homomorphism of
M l
into
M l
There is the injection monomorphism Horn ((M , Ml)I(j') l
E~~(FMI/F) la
a
of
into the endomorphism ring
of the linear space
The elements
.
of
has determinant
SFU Horn ((M l , Ml)/~)
FM
over
l
U Horn (M , Ml)ler) l 1
F
for which
form a normal subgroup with abelian factor group.
The er-equivalence classes pertaining to the
M '-S 2
are in one-to-one correspondence with the right cosets of
U Horn (rfrMl, cfMl);(j)
U Horn ((M , M ) I l l
fr)
over
U
whereas the proper
er-equivalence classes are in one-to-one correspondence with the right cosets of
If
.,.
=
LZ
then the number
er-equivalence classes of the
h (Ml/(J)
M 's 2
of
is finite
FINITE GROUPS
190 (see [8]).
~equivalence
The number of proper
classes is of the form
A)
U Horn ((M , Ml)lb) = S([)U Horn ((M , Ml)liJ) l l
((~Ml' (jM l ) lifJ =
U Horn
Et.ID U Horn
(tD M1 ' D-M 1 ) I (j1
or
B)
((M , M ) l(fl l l
U Horn
= It is
2
C)
2
•
if U Horn ((M , M ) Ibf l l U Horn (!jMl,OM
So
l
Horn ((M , M ) 16) l l
)tO'1 : -
-
(I),
(II),
-
U Horn ('bMl' D'M l ) 10')
Under the conditions
THEOREM 3 [9].
= ~U
The
".-modules
= 2
(Ill)
M , M l 2
are
properly er-equivalent if and only if the
1i
localization l'-;i Ml
of
Ml
we have
is properly
191
li
o;i-eqUiValent to the
o-'iM2
(i
= 1,2, ... ,s)
{~I
O"""i ()1 i
"1 i
0.,
-localization
when we set SE: 1)',
S
~'i}
,
cr·
The proof uses a simultaneous unit approximation theorem which is interesting on its own account since similar approximation theorems turn up decisively in the theory of quadratic forms (s.
[5]) and algebraic groups (s. Denoting by
h
1
~-eqUiValence classes distributed (for
,
[4]).
the number of proper into which the
~J
M2 's
are
any prime ideal) we obtain the
formula h + (Ml/V)
(5)
,
IT
hi
(#,
Ml/~J 0-)
prime ideal of -&,.
analogous to certain Siegel formulae (s. e. g. N.B.
1.
(1
<
i
<
h,(., M l/"10')
s)
.
is
1
if
1 I'i
[6]).
192
FINITE GROUPS
REFERENCES 1.
Arnitsur, Avraham Shimshon, Finite subgroups of division rings, Trans. Amer. Math. Soc. 80 (1955), 361-386.
2.
Chevalley, Claude, L'arithmetique dans les algebres de matrices, Act. Sc. et Ind. 323, Hermann (1936).
3.
Deuring, Max, Algebreni Ergebnisse der Mathematik und Ihrer Grenzgebiete IV NR 1, Springer (1935).
4.
Kneser, Martin, Starke Approximation in Algebraischen Gruppen I, Crelle J. Reine U. Angew. Math. 218 (1965), 190-203.
5.
O'Meara, Tim, Introduction to Quadratic Forms, Springer-Verlag (1963).
6.
Siegel, Carl Ludwig, Symplectic geometry, Amer. J. of Math. 65 (1943), 1-84. p. 6 Theorem 12.
See e.g.
193 7.
Steinitz, E., Rechteckige Systeme und
Modu1~
in A1gebraischen Zah1korpern I, II, Math. 71 (1911) 8.
328-354; 72 (1912)
A~~.
297-345.
Zassenhaus, Hans, Neuer Beweis der End1ichkeit der K1assen Zah1 Bei Unimodu1arer Aquiva1enz Enducher Ganzzah1iger Substitutions-gruppen, Abh. Math. Sem. Hamburg 12 (1938), 276-288.
9.
Zassenhaus, Hans, Aquiva1enz Ganzzah1iger Darste11ungen; Nachr. der Akademie Gottingen, Mathe Mathematisch-Physika1ische K1asse, Nr. 12 (1967) 167-193.
THE ORDERS OF THE FINITE CHEVALLEY AND STEINBERG GROUPS
R. W. carter
We consider the equivalence of the additive and mUltiplicative formulae for the orders of the Chevalley and Steinberg groups over finite fields. Let Y K
be a complex simple Lie algebra and
be the field
simple group
Then the order of the
GF(q)
G = -Se(K)
associated with
Y, K
by
Chevalley [1] is given by the multiplicative formula mQ,+l 1) ••• (q
where
Q,
is the rank of -Se,
posi ti ve roots of determined by m , ... ,mQ, l
Y
Y and
and
d
K.
are determined by 195
m
-
1)
is the number of
is a certain integer The integers .5£. only and satisfy
196
FINITE GROUPS
the condi tion
They may be defined in various alternative ways. They were introduced by Chevalley as the integers such that the Poincare polynomial of a compact Lie
Y
group of type
(t
is
2m +l 2m +l 2m +1 l - 1) (t 2 - 1)··· (t £
-
1)
Using the Bruhat decomposition a canonical form is obtained for elements of
G
which gives
rise to the additive formula
1:. d
where
qm(q _ 1)£
L
q
£ (w)
WEW
W is the Weyl group of
Y
and
£
(w)
the number of positive roots transformed by into negative roots.
is w
In order to prove the
equivalence of the additive and multiplicative formulae one must show
L WEW
q
£ (w)
=
(q -
1) £
Chevalley proves this by considering a cellular
197 decomposition of the coset space of a compact Lie group of type
Si' with respect to a maximal torus
of the group and computing the Poincare polynomial. More recently Solomon [2] has pointed out that this equivalence can be derived by considering only the Weyl group of
Si', and that a more general
formula can be proved valid for arbitrary Euclidean reflection groups. ml,···,m t under
In Solomon's work the integers
are defined in terms of the invariants
W of the algebras of real valued poly-
nomials on the Euclidean space on which
W acts.
The invariants form a subalgebra generated by the unit element and
n
algebraically independent
homogeneous polynomials of degrees m + 1, m + 1, ... , m + 1 . 2 t l Even in Solomon's proof a topological argument is still needed, and we therefore suggest another proof of the equivalence of the additive and mUltiplicative formulae, using only a knowledge of the root systems of the simple Lie algebras and the orders of their Weyl groups. simply define the integers for each type of algebra.
ml, ... ,m
We shall t
directly
They are as follows:
198
FINITE GROUPS
Y
~l'
..• ,m£,
A£,
1, 2, 3,
B£,
1, 3, 5,
C£,
1, 3 , 5,
D£,
1, 3, 5,
G
1, 5
F E E E
2
· .. , · .. , · .. , · .. ,
£, 2Q,
-
2Q,
- 1
2Q,
-
1
-
3, £,
1
1, 5, 7, 11
4
1, 4, 5, 7 , S, 11
6
1, 5, 7 , 9, 11, 13, 17
7
1, 7, 11, 13, 17, 19, 23, 29
S
The order of
W
is given in terms of
.
ml, ... ,m£,
by (m l + 1) (m 2 + 1)··· (m£, + 1)
IWI
We prove the equivalence of the two formulae by induction on the rank generated by
£,
£,
of
Y.
elements of order
the fundamental reflections of
Y.
W 2
is
which are It is possible
to omit one of these reflections in such a way that the subgroup
Wo
generated by the remainder
is the Wey1 group of a simple Lie algebra of rank £, - 1 .
Each coset of
Wo
in
W contains a
"distinguished coset representative," i.e. an
199
element
w
such that
w e WOw
with
¥
w
w.
this coset has form 9,(w)
=
9,(w)
9,(w)
for all
In fact each element
= wOw-
w
9, (wO) + 9,(w)
<
•
Let
where D
Wo e Wo
q
.Q, (w)
I weW
weW We know
I
q
9,
(w)
q
9,
(w)
of
and
be the set of
distinguished coset representatives.
L:
w
'i'
Then
q
• L
9,
(w)
.
WeD
o
by induction, so provided we
weW o can evaluate 2!xpression for
I weD
q
9,
I WEW
(w)
q
9,
we obtain the desired
(w)
•
Now the lengths of the distinguished coset representatives can be found rather easily by considering the root systems of the simple Lie algebras.
We use the following formulae:
IW(A9,)
W(A9,_l)
IW(B9,)
W(B9,_l)
IW (D 9,)
W(D9,_l)
IW(G 2 )
I W(B 3 ) I W(D S ) I W(E 6 ) I W(E 7 ) I
/W(F 4 )
IW(E 6 ) IW(E 7 ) IW(E a)
W(A l )
I I I
= 9, + 1
m(A
= 2.Q,
m(B9,+l) - m(B .Q, ) -
= 2.Q,
m(D
=
6
Hl
Hl = m(G ) 2
= 24
m(F 4)
= 27
m(E 7 )
= 56 = 240
m(E ) S = 2m(E ) S
)
m(A9,)
) - m(D.Q,)
m(E ) 6 m(E ) - 1 7
.
FINITE GROUPS
200
These formulae show that we can obtain information ~
about the coset representatives by embeading in a simple Lie algebra ~
£ + 1
of rank
when
this is possible, and considering the set Y
of
positive roots of the larger algebra which are not roots of the smaller one. permutation group on
W operates as a
Y.
W is transitive on
unless there is a positive root of ~ to all roots of
~.
orthogonal
If there is such a root,
this root is invariant under
W
and
is
W
transitive on the remaining roots in
Y.
be the highest root in this orbit of
W
the stabilizer of
R
Y
in
W
is
Wo
.
.
Let
R
Then
Also i f
w
is a distinguished coset representative one can show that
t(w)
=
where
h
h(R) - h(w(R))
or
h(R) - h(W(R)) - 1 ,
is the height of a root.
Since the
heights may be obtained from a knowledge of the root system the lengths of the distinguished coset representatives can be calculated. The method outlined here does not work if ~ cannot be embedded naturally in a larger Lie
201
algebra, viz if
Y
is
G2 , F 4
or
ES .
However
in these cases the required information can be obtained from the root system of W(E ) S
Y
itself.
is transitive on the roots of
stabilizer
W(E ) . 7
W(F 4 )
and
ES
W(G 2 )
with
have two
orbits, but the stabilizer of either orbit has the required form.
t(w)
=
where
R
One shows in all cases that
h(R) - h(w(R»
or
h(R) - h(w(R»
is the highest root of
Y
and
- 1
w(R)
an
arbitrary root. The method can also be adapted to give a proof of the equivalence of the additive and multiplicative formulae for the orders of the finite Steinberg groups.
REFERENCES 1.
C. Chevalley, Sur certains groups simples, Tohoku Math. Jour. 7 (1955), 14-66.
2.
L. Solomon, The orders of the finite Chevalley groups, Journal of Algebra 3 (1966), 376-393.
ON SOME PROPERTIES OF GROUPS WITH BN-PAIRS Nagayoshi Iwahori
1.
This note is a continuation of [2]
with several remarks. elsewhere.
together
Detailed proofs will appear
In order to state our main result and
its applications, we have to prepare several notations and concepts. Let
G
be a finite group and let
B, N
subgroups of
G
such that the pair
a BN-pair in
G
in the sense of J. Tits [4].
W be the Weyl group of this BN-pair: and let
R
+
W
forms
N/(B
TI
n N)
the canonical projection
W , one knows [3] that R
Let
be the distinguished generator set of
W , i.e. denoting by N
(B, N)
be
{n(O)i
o s N,
subgroup of
B U BoB
204
FINITE GROUPS
Every element of subset
of
J
R
,
R
generated by
W
of
is involutive [3).
denote by
generated by
G
and
B
[3].
~ GJ
J
subgroups
P
above.
R
of
THEOREM 1.
J
-1
Let
Now let
G,
Denote by
Furthermore the association
G
such that
k
P
k
J
(resp.
P
J
(resp. of
the number
0
J
W IG
)
2
,
0
J
the
( ex)
G
and
the k-represen-
induced by the of
r
G
J
(resp.
of inequivalent irreequals with that
PJ == mlP l +
let
JJ == n 1 0 1 + J
be as
, (ii)
2
B .
is a splitting field for
ducible constituents of 0
of all
'I'hen
).
(i)
of
~
R
does not divide the order
J
J
2
_9(G, B)
be a field such that
trivial representation W
Then
G, B, N, W, R, J, W , G J J
k
( B)
tations of
of
the subgroup
(W J)
onto the set
G
characteristic of
W.
IT
G
is a bijective mapping from the set
of all subsets of
of
the subgroup of
W J
Denote by
J
Given a
+ n r
°r
...
+ mrPr
,
be the decomposition of
into inequivalent irreducible constituents
1 " , "P r
i
01'"
.,0 r
with multiplicities
205
ml,· .. ,m r
;
nl, ... ,n r
respectively.
a suitable ordering of the n.
m.
< i
(1
1
1
Then under
Pi' we have
< r)
This theorem is of course a natural generalization of a theorem of R. Steinberg [5] for the case
G
=
Although Steinberg
GL (n, :IF )
q
has given a more precise information about the degrees of the
p.,
O.
1
]
in his case, we don't
know how to generalize it to the general case.
We
should also remark that Theorem 1 is proved by J. Tits as is sketched in [2]
empty, i.e.
G
J
= B,
W
for the case
J
is
{l} ; and that our
J
proof is based still on the main lemma below due to Tits which played a basic role in the proof in [ 2] •
LEMMA 1 (Tits).
Let
er
be an integral domain
with the unit element and let field of over
er (i)
er.
A
Let
K
be the quotient
be an associative algebra
such that A
is a free
er-module of finite rank,
and (ii)
A = A 0 K
er
K
is a separable algebra
206 over
FINITE GROUPS K. Let
k
unitary ring homomorphism. ~-module
~
be a field and let
by
=
a·x
=
e~
Regarding
~(a)x A~
the tensor product
:
(a A
~e
E: (!j,
k X
k
be a
as an E:
k)
, form
Then
k .
can
be regarded as an associative algebra over
k
in
the usual way. Suppose that over of
k. K, k
is a separable algebra
A~
Denote by
K, k
respectively.
the algebraic closures
Let
A~ ~k k be the decomposition of the semi-simple algebras AK ~K K,
A~ ~k
k
simple components.
K, k
over
M (L)
(Here
v
f:1atric algebra of degree Then
r = s
and
v
m. l
a suitable ordering of the ~emark.
respectively into means the total
n. l
.)
L
over (1
<
=
i
<
r)
for
m. l
Although the proof of this lemma
sketched in [2] was based on the rigidity property of deformations of separable algebras, an elementary proof was given by J. Tits.
207
Our proof of Theorem 1 is qUite similar as the case of Tits in [2]. that M'k(G, G ) " J
It is enough to show
M'k(W, WJ )
Hecke algebra over
where M'k
means the
in the sense of [2].
k
We begin with the following
(i)
LEMMA 2.
=
W* J
9, (T)
( cL
R
for every
> 9,(0)
Then
9,
Y E W* J
R.
be a subset of
is the length of
[3] ) •
,
x E W J
J
9,(ro)
{o E W
where
Let
(xy)
=
T E W 9,
(x) +
Furthermore
W
r
Put
E J}
relative to (y)
for
= U
WJo
9,
OEW * J
is
a disjoint union. (ii) G(J -
l'
Let
J
J ) ~ W* 2
J
U
W
oEG(J ,J ) l 2 Now let
n
l
, J
2
(w* )-1 J
l
c
;
We write by A
er
u
be the set of conjugacy classes
for
r
u
c
R, and let
in case
the polynomial ring
be the free
indexed by
Then
be indeterrninates over the field
E C}
C
Put
is a disjoint union.
represented by elements in {u
R.
2
W OW J J 1 2 C
be subsets of
W
k[u
r E c c
i
.
bijectively as its base.
.
Denote
c E C)
er-module with the set
k
Let
{aD ; a E W} Then one
FINITE GROUPS
208 knows [2] of
that there exists uniquely a structure
er-algebra in the
r s R,
er-module
A
such that for
w s· W
if
a
a a r
w
u::r~ +
{
9,
9,
(rw)
> 9,(w)
if
(u r - l)a w '
(rw)
< 9, (w)
Furthermore one has
LEMMA 3.
As an
tar ; r s R}
l
2
is generated by
= ar
(for every
r s R)
u r 'a 1 + (u r - l)a r
(for every
r s R)
a a r
=
1
a a a a r s r s
-v---' m
A
together with the defining relations
alar ar
er-algebra
(for every
a a a a'" s r s r
~
in
m r,s
r,s
R
I
m
rs).
Hence one can verify easily that there exists
such that Define now
er-algebra homomorphism v(a )
for every
r
sJ sA,
d
J
s
er
r,s
being the order of
uniquely an
r,s
r
v in
as follows:
R
209
a
o
Then one can show that u d
d
OSW
.
0 J ~
Define now an {x
Then
A
{e o·
elements
Hence
J
6'-submodule
A
J
of
A
by
A
S
is a free
J
£(O)
Max
of degree equal to
c
is a polynomial in the
J
6'-module spanned by the
o S 8(J, J}}
I
a
where
T
Hence by introducing a
*
new multiplication (x
S
A
J
Y
I
S
A )
6'
~
k
we can show that
by
x
*
Y
= ~ d
(xy)
J
6'-algebra
be k-algebra homo-
defined by
respectively, where Bx
J
ep ,1/1
(r S
of the form
A
, we get an
J
Le t
morphisms
in
qr
R)
, 1/1 (u ) = 1 r
is the number of
contained in
BTI
-1
(r}B.
(r
S
R)
B-coset~
Then
210
FINITE GROUPS
Thus finally we get Theorem 1 by Lemma 1.
Let
THEOREM 2. 0
J
,
ml, ... ,m r
G, E, N, W, R, J, G , W , P , J J J
' nl, ... ,n r
be as in Theorem 1.
Suppose that every complex representation of is realized over the real number field. following conditions for a subset
Then the
of
J
W
Rare
all equivalent. is commutative,
(1 )
(2)
is commutative, (4)
n
1
=
for every
n
x
1
r
in
G, for every
( 6) (7 )
(5 )
every element of
8(J, J)
The equivalence of (1)
~
0
in
W,
is involutive. (6) is easy from
Theorem 1 and a theorem of J. S. Frame [1]. (6) is equivalent to (7) by Lemma 2. using the criterion (7) above, one can give a counter example to the following conjecture: if
G
J
is a maximal subgroup of
is commutative. (\.;,
R)
~~(G,
G,
G ) J
Namely consider the case where
is of type
(D S ) , e.g. let
G
be the
special orthogonal group on ten variables over a
711 finite field with respect to the quadratic form 10
L
X.X
i==l
1.
lO
_·
, and let
B
be the sUbgroup consisting
1.
of triangular matrices in R == {r , r , r , r 4 , r } l 3 S 2 r.r. 1.
is
3
or
2
J
We have then
G
m ..
and the order
according to
r.
and
l
of
1.J
r.
is
J
connected or not by a segment in the diagram below:
Let
== R -
J
element
{r } . 3
Then
r3r2rlr4r3rSr2r3
G(J, J)
contains the
which is not involutive.
Similarly a counter example exists for the type (F 4)
(The conjecture above is verified to be
true for types
(A ),
n
(B ),
n
(C ),
n
(G
2
)
.)
REFERENCES 1.
J. S. Frame, The double cosets of a finite group, Bull. AIDer. Math. Soc., 47, 4S8-467 (1941) .
2.
N. Iwahori, Appendix of generalized Tits system (Bruhat Decomposition) on
p -adic
212
FINITE GROUPS semi-simple groups, Proc. of Symposia in Pure Mathematics, 9 (1965).
3.
J. Tits, Theoreme de Bruhat et sous-groupes
paraboliques, C. R. Acad. Sci. 254 (1962), 3419-3422. 4.
, Algebraic and abstract simple
groups, Ann. of Math., 80 (1964), 313-329. 5.
R. Steinberg, A geometric approach to the representations of the full linear group over a Galois field, Trans. Amer. Math. Soc. 71 (1951), 274-282.
University of Tokyo
THE STEINBERG CHARACTER OF A FINITE GROUP WITH BN-PAIR Louis Solomon*
Let
1. field
F
G(q)
be a Chevalley group over the
Steinberg [3J showed that
q
G(q)
has
a remarkable irreducible character of degree equal to a power of
q
any finite group character G
~
G(q)
X
Later, Curtis [lJ showed that G
with BN-pair has an irreducible
which is equal to Steinberg's in case
, and that
X
may be written as an
alternating linear combination of characters induced from parabolic subgroups of
G.
In view
of [2, §4] one might conjecture that this formula has a homological source. is the case.
We show here that this
The requisite complex has been
defined by Tits [5, 6].
I owe a great debt to
* This work was supported in part by the National Science Foundation under grant GP-6080. 213
214
FINITE GROUPS
Tits for the lesson in "Anschauliche Geometrie" which led to the present argument.
2.
Let
rank
G
be a finite group with BN-pair [4] 1
and let
subgroups of
G , ••. , G
G
1 = VU···
U V
Q,
gG
.
i
s
V
v
is nonempty.
l G , ... ,GQ, s E 6 on
6
of dimension
6
Q,
-
g E G
G
1
A collection
if and only if
A simplex of dimension
Q,
-
1
The chamber with vertices
then
gs E 6 .
If
This action of
preserves the simplicial structure and
thus induces an action of groups of
be the
is called the fundamental chamber.
and 6
vi
g E G , and let
as its set of vertices.
is called a chamber.
G
,
of vertices is a simplex of
n VES
Let
B
The Tits complex [5, 6] of
is a simplicial complex which has
be the maximal parabolic
containing
collection of cosets V
Q,
of
G
on the homology
6
THEOREM 1.
Let
finite group
G
be the Tits complex of a
6
with BN-pair of rank
homology groups of
6
Q,
> 2
The
with integral coefficients arE
215
H.(6)=O
1
l
H (6) £-1
z e'"
~
t
Z
$
involution of the Weyl group
is the
I";
=
modulo
If
bl, ... ,b WOBW O n B
t
THEOREM 2.
Let
finite group
G
on
Q 6
then
6
a
CtL
-+
is the
£ - 1
chain
defines a
bll";, ... ,btl";
are
H£_l (6)
be the Tits complex of a
with BN-pair of rank
be the rational field.
H£_1(6) 0 Q of
W
represent the cosets
cycles which form a basis for
let
E
Wand
then the
oppositio~
LWE W E (w) wa
is a cycle. B
6
If
W
is the alternating character of fundamental chamber of
£ - 2
~
i
summands
and
where
of
<
£
~
2
and
The action of
G
Q[G]-module structure in
which affords the Steinberg character
G.
3. here.
Space does not permit a complete argument We try to explain why one should expect
such theorems and sketch the proof. point is that one should view
The main
W as a degenerate
216
FINITE GROUPS
form of
G
and hence theorems about
W should
appear as degenerate forms of theorems about Let
1 W, ... ,W £
of
W.
by
1 W, ... ,W £
by
G , ••• , G
be the maximal parabolic subgroups
The Coxeter complex
1
G
of
I::
W
in the same way that
£
is defined is defined
One knows that there is a
faithful representation
p
of
W in the
orthogonal group of a Euclidean space
R£ , in
which the distinguished involutory generators rl, ... ,r£ Let
of
Ware represented by reflections.
be the simplicial complex cut out on the
1::'
unit sphere
£-1
S
by the hyperplanes corresponding
to the reflections in
e
isomorphism such that
e
0
I:: +
w
p(W)
of simplicial complexes
1::'
p(w)
There exists an
0
e
for
Thus
has the homology of a sphere which means, since
I::
£ > 2 , that i
HO (I::) " Z ,
1, .. .,£ - 1 , and
H£_l
Hi
0
w
=
p(w)
0
e
for
the representation of
(I::)
(1::)
degenerate form of Theorem 1.
e
w E W.
0
"Z
for This is a
Since
w E W , the character of W on
H£_l(l::)
is equal
to the character of the representation of
W on
and hence, by an argument given in [2, §4] is the alternating character of
W.
2]7
A glance at [3, Theorem 1] makes it apparent that this fact is a degenerate form of Theorem 2. These observations lead one to conjecture the truth of Theorems 1 and 2 on the strength of known
W.
facts about
In a conversation at Oberwolfach, Tits suggested that one could prove Theorem 1 as follows.
Follow the analogy in the preceding
paragraph.
To compute the homology of
one can choose a point
p E S£-l , let
S
9,-1
p'
antipodal point, and prove that
S£-l - {p'}
contractible by deforming it to
p
circles.
be the is
along great
Thus one should delete the chambers which are, in a precise sense,
of
opposite the fundamental chamber and prove that D - {Cl' ... ,c } is contractible. This is the t crux of the idea in the present argument although the formal proof contains less geometry and more combinatorics.
LEMMA 1.
Let
is a union
K
Suppose point,
K
be a simplicial complex which
LULl U ... U L
of subcomplexes. n has the homology of a
(i)
each
L.
(i i)
each
L n L.
l
l
has the homology of a
FINITE GROUPS
218
point, and Then
K
(iii)
and
Let
c
s
c
n c'
d(c)
let
such that
c
i
w r
LEMMA 2.
If
( i)
and
c. l
If
1
~
(ii)
If
w E W
c E r
Let
then
c ' c + i i l
cO,c1,···,c n
let
£(w)
be the
sEt!..
c E
r (s)
and
c
for some
Then
E
Co E r(s)
such that
r (s)
d(c) = d (co) + n
co,cl'···,c n = c
are adjacent and
i=O, ... , n - l .
= bwa
c
= £(w)
d(c)
for all
d (c)
there exist
=
are adjacent for
There exists a unique
d(c O)
a
, ... ,r£ .
w E W , and
LEHMA 3.
If
as a word in the distinguished
generators
b E B,
.
be the least nonnegative integer
O, ... , n - l
length of
are
£ - 2
has dimension
t!.
c E r
c, c' E r
for which there exist chambers
n
=
r
denote the set of Say that
;;;2
adjacent if E
r (s)
let
such that
c
have isomorphic homology groups.
denote the set of chambers of
r
s E t!.
If
L
E
r (s)
d(c ) i
=
then
such that d(c O) + i
for
219
Lemma 2 follows from a theorem of Tits [5, proposition 2].
Lemma 3 requires Lemma 2 and
the distinguished coset representatives [2, §3] for
W modulo the parabolic sUbgroup determined
naturally by
s.
r
ranges over set of with
Let
s E 6
If
0 < k < m
Then
=
6k- l ~ 6k
and
If
o
where
< k
<
m
c
let
6
c
be the
k
for some
q, (s)
let
~
P
Thus
~
<
q,(c)
1
-
n
6
be the
and suppose
Let
d(c)
=
be the
c
k ~
- 2
numbered sO that Then
sl'··· ,sp E 6 k _ 1 1
r
c E
and its faces.
s
dimensional faces of
where
is a subcomplex of
s E 6
c E r
Let
~
s 6k
complex consisting of
LEMMA 4.
max d(c)
such that
< k
d (c)
m
and 6 k l
has the homology of a point.
The critical ingredient in the proof of Lemma 4 is Lemma 3 (ii). 0 < k < m
of
6
L
6k - 1
.
Let
for which and let
Now choose some
l
L. l
=
=
wit~
be all the chambers
al,···,a n d (a.)
k
k
Let
q, (a. ) l
K
=
6 k , let
Lemmas 3
(i)
and
4 allow us to apply Lemma 1 in this situation and
,
220
FINITE GROUPS
we conclude that
6
and
homology groups.
Thus
k
6 k l
6m-
have isomorphic
and
l
6
0
have
isomorphic homology groups which means that has the homology of a point.
If
all the chambers of
d(c.)
6 m-1
i
=
6
Since
is connected and thus H -
l
t
chambers
(6)
are
t
~
H (6) O
= m then
1
and thus
{c 1 ' ... , c t }
6 -
= l, ... ,t - 2.
about
with
6 m l
H. (6) 1
= 0
for
2 , the complex ~
Z.
6
The assertions
depend on the fact that the
biwOa,
i
=
l, ... ,t
are precisely the
chambers As for Theorem 2, one argues as in [2J using Theorem 1 and the Hopf trace formula.
This gives
an expression for the character of the representation of
G
on
H t
l
(6) 0 Q
as an alternating sum
of induced characters and this sum is precisely the expression for the Steinberg character which Curtis has given in [lJ.
REFERENCES 1.
C. W. Curtis, On the Steinberg character of a finite group with 4
(1966),
433-441.
(B, N)-pair, J. Algebra
221 2.
L. Solomon, The orders of the finite Chevalley groups, J. Algebra 3 (1966), 376-393.
3.
R. Steinberg, Prime power representations of finite linear groups (11), Canadian J. Math. 9 (1957), 347-351.
4.
J. Tits, Theoreme de Bruhat et sous-groupes parabo1iques, C. R. Acad. Sci. Paris 254 (1962), 2910-2912.
5.
, Structures et groupes de Weyl, Seminaire Bourbaki, 1965, Expose 288.
6.
, manuscript to appear in the Springer series of Lecture Notes.
New Mexico State University
A SUFFICIENT CONDITION FOR p-STABILITY George Glauberman
Let p-group
p
be an odd prime.
S , let
d(S)
For every finite
be the maximum of the
orders of the abelian subgroups of
S.
the Thompson subgroup
to be the
J(S)
of
S
Define
subgroup generated by all the abelian subgroups of order Let
d(S) E
in
S
be a two-dimensional vector space
over the field of
p
elements, and let
the special linear group on
SL(E)
be
E , i.e., the group
of all linear transformations of determinant one on
E.
Define the quadratic group
the semi-direct product of F(p)
E
by
Qd(p) SL(E)
to be Let
be the normalizer of a Sylow p-subgroup of
Qd(p) A group
H
is said to be involved in a group 223
224
G
FINITE GROUPS
if
H
~
K/L
such that
for some subgroups
L
is normal in
Now suppose finite group S.
G.
Sand c
X
g
E
G
E
CG(X)
T
G
K.
T
be a normal sUbgroup of
controls strong fusion in
and
and
of
is a Sylow p-subgroup of a
Let
We say that
if, whenever
some
S
K, L
Y
are non-empty subsets of g X
Y
G
and some
then n
E
g
cn
NG(T)
for
It is easy
to see that this definition does not depend on the choice of
S
and that it corresponds to the
definition given by Alperin and Gorenstein [1]. Recently, the following results were obtained [ 2] :
Z(J(S))
G
If
N(Z(J(S)))
G, then
G
has a normal p-complement,
has a normal p-complement.
Let if
is not involved in
controls strong fusion in
(2)
then
Qd (p)
If
( 1)
N(Z)
Z
=
Z(J(S))
It follows from (2)
has a normal p-complement, then
is not involved in
G.
that
Qd(p)
The main result of this
paper is a generalization of this observation.
THEOREM A.
If
F(p)
is not involved in
N(Z)
225 then
Qd(p)
is not involved in
G .
By (1), we obtain
COROLLARY 1. N(Z)
, then
If
Z
is not involved in
F (p)
controls strong fusion in
G
Now (2) follows from Corollary 1 and an easy transfer argument.
is involved in
F(p)
COROLLARY 2.
only if it is involved in
G
if and
N(Z)
Theorem A follows from:
Let
THEOREM B.
normal subgroup of
Suppose
N(J(SJ")
H =
H
contained ln
does not control strong fusion in exist subgroups
V
and
V
of
Z , and
with the
following properties: (i) (ii) (iii) (iv)
V
is normal in
V/V (V
NG(U)
automorphisms of
V
is elementary of order
n T)/U
has order
n NG(V) V/U
p
i
p2
and
induces a group of
that contains
is a T
Then there
G
S
T
SL(V/U)
226
FINITE GROUPS Moreover,
COROLLARY 3.
F(p)
is involved in
N(T)
Assume the hypothesis of Theorem B.
Then
T
contains a subgroup
that
NH(C)/CH(C)
C
of order
is cyclic of order
p
such
p - 1
REFERENCES 1.
J. L. Alperin and D. Gorenstein, Transfer
and fusion in finite groups, J. of Alg. 6
2.
(1967),
242-255.
G. Glauberman, A characteristic subgroup of a p-stable group, Can. J. Math., in press.
ON THE CENTRALIZERS OF INVOLUTIONS IN FINITE GROUPS Daniel Gorenstein
In each of the known simple groups the centralizer
N
of any involution
restricted structure. to be the case that that
C (0 (N) ) N
x
has a very
In particular, it appears O(N)
is always cyclic and
has index at most
2
ln
N
The question of whether some result of this nature holds more generally for arbitrary simple groups, at least for certain involutions
x , is of
paramount importance for the ultimate classification of the finite simple groups.
A major
portion of the solution of every general classification problem considered to date has involved the successful resolution of this problem.
The
general question is itself very closely related to certain questions raised by Thompson concerning
227
228
FINITE GROUPS
2-signalizers of simple groups (that is, the set of subgroups of odd order normalized by some Sylow 2-subgroup of the given simple group). We have recently obtained a theorem which represents a seemingly necessary step toward the solution of the overall problem.
To achieve the
broadest scope for possible future applications, we have deliberately stated our result in very general terms.
However, there is strong reason to
believe that this degree of generality will, in fact, be required to obtain a complete analysis of the structure of
O(N)
and of its embedding in
N
To state our theorem precisely, we need several preliminary notions, which we proceed to define.
DEFINITION 1. and let m(A)
~
A
Let
G
be an abelian 2-subgroup of We shall call
3
functor on
be a group of even order
G
e
provided for each involution e(CG(u))
with the following properties: and
( a)
e(cG(u))
<1
with
an A-signalizer
A , there is associated a subgroup CG(u)
G
u
of of
229
involution (c) m(A/B)
~
v If
of B
A is a subgroup of
A
such that
2 , then x
e(cG(u » for all involutions
u
in
B
and all
x
in
These conditions are obviously satisfied if e(cG(u»
= I
for all involutions
this case we call functor. for all
e
in
e(cG(u»
is of particular importance. e = O.
case we set
A.
In
the trivial A-signalizer
The special case that u
u
= O(CG(u»
In this
Actually it can be shown
that a somewhat weaker assumption than the normality condition in (a) SUffices for our main results.
DEFINITION 2.
Let
functor on
We denote by
G
A-invariant subgroups that
e
be an A-signalizer
K
of
He (A) G
the set of
of odd order such
FINITE GROUPS
230
n 8(C G (u))
K
u
involutions of As usual, we write and
K
A > .
1-1 8 (A;
p)
if
111 8 (A)
K s
is a p-group.
If Ki s
K =
ranging over the
K
H 8 (A)
=
I 1 ~ i ~ m>
and each
, then clearly also
K s
H 8 (A) X
Furthermore, because of condition (c), whenever
K s
H 8 (A)
and
K
s
H 8 (A)
In addition,
x s NG (A)
it is a consequence of the definition that if K s ~8(A)
, then
involution
u
of
CK(u) c::: 8(C (U)) G A
But then if
A-invariant sUbgroup of that
L s
V1 8 (A)
for any L
is an
K , it follows at once
These basic properties of
8
are used repeatedly. If for all involutions
u
of
A,
O(CG(u))
is the unique maximal A-invariant subgroup of C (u) G
, then it is easy to see that
A-signalizer functor on of
V\ (A)
~O (A)
G
is contained in
in this case.
8
=
0
and that every element
Vl o (A)
Thus
V\ (A)
However, in general,
will be a proper subset of V\(A) can occur even in the case
is an
8
=
H 8 (A)
; moreover, this
= 0 .
231
e
DEFINITION 3.
Let
functor on
A subgroup
A
G
be an A-signa1izer H
of
G
containing
will be called A-flat with respect to
provided the elements of all lie in
~8(A)
e
contained in
H
O(H)
It is immediate from the definition if H = CG(u) fact,
H
u
an involution of
e.
is A-flat with respect to
DEFINITION 4. G
A, that, in
An A-signalizer functor
e
on
will be called flat provided any p-local sub-
group of
G,
P
with respect to
odd, containing
A
is A-flat
e .
In terms of these concepts we can now state our main result:
THEOREM A. the group
If G
with
A
is an abelian 2-subgroup of m(A)
~
6
the flat A-signalizer functor elements of
~8(A)
and
e ,
G
possesses
then the
generate a subgroup of
odd order invariant under
G
0:
NG(A)
We should like to explain the
relationshi~
232
FINITE GROUPS
of Theorem A to the general problem described above of determining the structure and embedding of
O(N)
in
N
This can best be done in the
context of some recent work of Thompson on groups of order relatively prime to
3 , [5]
, together
with his earlier classification of N-groups [4], in both of which a result similar to Theorem A appears. In the course of his study of 3 '-groups, Thompson has obtained what amounts to a very general sufficient condition for
0
itself to be
a flat A-signalizer functor of the group where
A
is an element of
2-subgroup of
G.
SCN(S)
S
G , a Sylow
His conditions involve
assumptions on the nonsolvable composition factors of the local subgroups of
G.
In particular,
these conditions are vacuously satisfied if an N-group and also hold if
G
G
is
is a 3'-group, in
which case the only nonsolvable composition factors to be considered are Suzuki groups. When with
A
in
G
is a simple N-group or 3 '-group
SCN (S) 3
, Thompson shows, using a
somewhat different approach from the one we use to prove Theorem A, that the set of all elements of
233
generate a subgroup order.
K
of
G
of odd
There appear to be considerable technical
difficulties involved in attempting a direct generalization of Thompson's procedure.
In part
at least, the significance of our Theorem A rests on the fact that its proof bypasses these difficulties.
Unfortunately our assumption
rather than
m(A)
~
3
m(A)
6 ,
~
as in Thompson's approach,
appears to be essential.
However, if an alternate
proof of One specific preliminary result of ours could be obtained, it would be possible to establish Theorem A under the weaker assumption m(A)
~
5.
However, this would seem to be as far
as one can go with our methods. Continuing his analysis of simple N-groups K t 1 ,
and 3 '-groups Thompson goes on to show, if that
M
=
NG(K)
is strongly embedded in
G , from
which he derives a contradiction by a now standard argument.
(See also Bender's talk, where a complete
classification of groups possessing a strongly embedded subgroup is presented.)
consequently the A-signalizer functor trivial.
K = 1
Thus 0
and
is
In Thompson's terminology, the effect of
his complete argument is to establish that
2
E
TI
4
234
FINITE GROUPS
Expressed in terms of our initial problem, his argument shows that if A , and
N = CG(x)
x
is any involution of O(N) = I
, then, in fact,
The preceding discussion should clearly indicate the role of Theorem A for the general problem: chain.
it is the middle link in a three-part The first part of the chain will involve
the existence of appropriate A-signalizer functors
e
on
G , based upon assumptions concerning the
nonsolvable composition factors of the local subgroups of
G
containing
A.
On the other
hand, the final link will involve a proof that
G
possesses no nontrivial flat A-signalizer functor
e ,
based upon the construction of a strongly
embedded subgroup K
M
of
G
in case the subgroup
generated by the elements of
~e (A)
is
nontrivial. One cannot expect to prove in general that
o
itself is an A-signalizer functor on
G, for
Thompson's conditions hold only under special circumstances.
In fact, they correspond primarily
to the case in which the nonsolvable composition factors of the local subgroups of
G
are of Lie
type defined over a field of characteristic
2.
235
Thus, for example, in the study of groups
G
witt
abelian Sylow 2-subgroups, recently classified by J. H. WaIter [6], Thompson's conditions are not satisfied and therefore functor on G.
G , where
0 A
is not an A-signalizer is a Sylow 2-subgroup of
On the other hand, if
m(A)
~
8 , one can
construct suitable flat A-signalizer functors on
G
E
and, using them, can follow the procedure
outlined above to obtain an alternate proof of WaIter's classification theorem (under the assumption
m(A)
~
8 ).
The construction of useful flat A-signalizer functors for the study of general simple groups appears to be extremely promising provided one takes
A
to be an abelian 2-subgroup of the given
simple group
such that
m(A)
is as large as
possible and provided also
m(A)
is sufficiently
large.
G
(Hopefully
m(A)
> 9
will suffice.)
We
are presently studying this problem with J. H. WaIter (see his talk). Finally a word about the proof of Theorem A. In spirit it conforms very closely to the conceptual framework of Chapter IV of the Odd order paper [1] as further extended and axiomatized
236
FINITE GROUPS
in [3], the primary objective being to establish appropriate E-theorems for the maximal elements of
Vie
(Ai
p
p)
ranging over all odd primes.
Because of our hypothesis on
e , the relation
J1 e (A)
between various elements of
can be studied
from a knowledge of the structure of u
ranging over the involutions of
e(cG(u» As a
A
result, the proof of Theorem A is considerably easier than the corresponding results in [1] and [3], even discounting simplifications due to Glauberman's recent
ZJ- and ZJ*-theorems [2],
both of which we use crucially. Of central importance to our argument is a so-called relativized transitivity theorem. P
be a maximal element of
lit e (Ai
p)
a minimal A-invariant subgroup of B
, let
Z(P)
be an elementary abelian subgroup of
with
m(B)
A/CA(Z)
4.
~
(Such subgroups
is cyclic and ~e
Also define with Me(A)
(B)
and
m(A)
~
6
lf1 e (Bi
and ltfe(Ai p)
Q
distinct from
of &.1
e (Bi
q)
Z
be
, and let CA(Z)
exist as
by hypothesis.)
p)
analogously
Then our transitivity
theorem deals with the set of subgroups
B
Let
B q
x
Z
invariant
an odd prime
p , which are maximal subject to
237 the condition
[Q, Z] = Q .
The proof of Theorem A does not depend in an essential way upon the fact that
A
is a 2-group.
As a result it has certain natural extensions to the case in which
A
is a p-group,
p
odd.
Thompson has pointed out that such results are also needed for the study of simple groups.
A
heuristic reason for this can be seen from the fact that in the groups of Lie type over fields of characteristic
2, the elements of odd prime
order in a Cartan subgroup have a structure analogous to those of the corresponding involutions in the groups of Lie type over fields of odd characteristic.
REFERENCES 1.
Feit,
w.
and J. G. Thompson, Solvability of
groups of odd order, Pac. Jour. Math. 13 (1963), 775-1029. 2.
Glauberman, J., A characteristic subgroup of a p-stable group (to appear in Can. Jour.).
238 3.
FINITE GROUPS Gorenstein, D. and J. H. Walter, On the maximal sUbgroups of finite simple groups, Jour. Alg. 1 (1964), 168-213.
4.
Thompson, J. G., Non-solvable finite groups all of whose local subgroups are solvable (to appear). Groups of order prime to
5.
3
(to
appear) . 6.
Walter, J. H., The characterization of finite groups with abelian Sylow 2-subgroups (to appear in Annals of Math.).
CENTRALIZERS OF INVOLUTIONS IN FINITE SIMPLE GROUPS John H. Walter
The purpose of this talk is to state some results about the centralizers of involutions in finite simple groups.
The research described is
not complete and thus the results are not in final form. A group
G
is said to be centrally simple if
it is a central product of centrally simple groups. group.
It is said to be centrally semisimple if
it is a central product of centrally simple groups. Let
O(H)
denote the maximal normal subgroup of
odd order of a finite group
H.
Denote by
L(H)
the maximal normal centrally semisimple subgroup A component of
L(H)
is a normal
centrally simple subgroup of
L(H)
; the Krull-
of
H/O(H)
Schmidt theory shows that the components are 239
FINITE GROUPS
240
uniquely determined. act on of
L(H)
H
Of course, subgroups of
In particular, if
disjoint from
its image in
H/O(H)
~.
Let
O(H)
K
H
is a subgroup
we identify. K
with
to simplify the notation.
denote the class of centrally
simple Chevalley groups together with perfect central extensions of the simple alternating groups.
~
Denote by
..
o
the class of those
centrally simple Chevalley groups defined over fields of odd characteristic. Now let
G
be a finite simple group.
that for every proper subgroup components of
L(H)
are in
elementary 2-subgroup of order.
where
subgroup of ~~.
LEMMA 1. A. of
of
G
the
Let
A
be an
of maximal possible
We are interested in the possibilities for
L(C(t))
in
G
~
H
Assume
t
E:
L(H)
A#
Set
LO(H)
to the
generated by those components
An important lemma is the following.
Let
H
be a subgroup of
G
containing
Then the minimal A-invariant normal subgroups LO(H)
are either centrally simple components
or the central product of two copies of q > 3
•
SL(2, q)
241
Now let Let EL L.
L
AO
be a nontrivial subgroup of
be a component of
= CA(L)
Then
Let ~
EL
LO(C(A )) O
Set
is said to be the base of
be the set of subgroups of
are bases for some component of
A
which
LO(C(A )) O
where
A O
runs through the set of nontrivial subgroups
of
A.
~
Then
is ordered by inclusion.
maximal component of
that two maximal components EL
THEOREM. let
A
Let
1
=2 A n L 2
G
L
and
l
and
B
L
is
We say L
2
A
~
A)
2
are
n Ll
be a finite simple group and
be an elementary 2-subgroup of maximal
possible order.
Assume that
that for each involution of
A
(with respect to
G
a component with a minimal base group.
disjoint if
A
L (C (t) )
are in
maximal component of belongs to
:Co
isomorphic to (1)
:c G
t
IAI of
> 2
6
Assume
A, the components
and that at least one with respect to
A
but does not have a simple factor PSL(2, q)
Then either
All maximal components are mutually
disjoint; or (2)
All minimal base groups are cyclic and
all maximal components are Chevalley groups
defi~ec
242
FINITE GROUPS
over fields of the same odd characteristic. This theorem has numerous and important applications.
Some of the techniques leading to
its proof are important in studying the construction Jf a signalizer functor as explained by Daniel Gorenstein in his talk.
The proof depends on
establishing a variety of properties for the Chevalley groups, and not all of this has been done in complete detail.
Nevertheless,
the
results have been investigated for most of the groups in question and at least apply to groups having general properties which can be conjectured for the entire class.
Thus to be more accurate
but less succinct, the class
~O
should be
defined in terms of these properties. It is expected that using the signalizer functors, case (1) may be eliminated and that in case (2), the centralizer of shown to cover (2)
L(C(t»
O(C(t»
can be
In this case Condition
leads to many important results, which will be
the subject of a forthcoming paper.
ON Y-PROJECTORS B. Huppert
Let
Y
be a sa tura ted formation (in the
sense of W. Gaschutz), an Y-projector of
G
G.
a soluble group and
F
(Y-projectors are the
generalized Carter-subgroups of Gaschutz.)
THEOREM 1. of
G
If
and
F
N l
an
and
N 2
~~-projector
are normal subgroups of
G, then
Combined with the trivial relation N l
n N2 n F
that
N
+
N
(N
n F
l
n F) n (N 2 n F)
this shows,
is a lattice-homomorphism of the
lattice of all normal subgroups of lattice of normal subgroups of
F.
G
into the
So there
exists in particular a unique normal subgroup 243
M
244
of
FINITE GROUPS G, maximal with respect to
M
n F
=
I
There seems to be no handy description of
M
Theorem I seems to be related to Wielandt's theorem on the lattice-homomorphism of the lattice of all subnormal sUbgroups of
G
into the lattice
of subgroups of a Sylow subgroup of
G
But
Theorem 1 becomes wrong for nearly all formations g
if "normal" is replaced by "subnormal".
Wielandt's results suggest the question under what conditions the lattice-homomorphism of Theorem 1 is an epimorphism.
Let
THEOREM 2.
g
be a saturated formation,
locally defined by the formations
g(p)
Assume
that one of the following assumptions holds: (a)
All~(p)
's
are equal.
(This covers for
instance the formations of nilpotent groups and groups with nilpotent commutator groups.) (b)
If
H
subgroup of is in
g
i
subgroups.
E
g
and
N
is a minimal normal
H , then the automorphism group of further
g
is closed under taking
(This covers supersolvable groups.)
Then the following statements are equivalent:
N
243
(1)
The lattice-homomorphism of Theorem 1 is an
epimorphism. (2)
If
denotes the g-residuum of
Gg
smallest normal subgroup of _~)
in G
=
and
F
and
GgF
If
G
(the
with factor group
anY-projector of Gy
G
G, then
-n F = 1 .
A, B, C
are subgroups of
G, then the
following statements are trivially equivalent: (1 )
A
(2 )
AB
n n
BC
(A
AC
n
B) (A
n
A(B
n
C)
C)
Using this we get from Theorem 1 a result of
J. Rose
(which now appears to be equivalent to
Theorem 1), namely:
THEOREM 3.
soluble group subgroups of
If G
F
is
and
an~-projector
N
l
and
N
of the
are normal
2
G, then
This means that the mapping
N
~
NF
is a lattice-
homomorphism of the lattice of normal sUbgroups of G
into the lattice of subgroups
U
with
246
FINITE GROUP S
F < U < G .
(Will appear in Archiv der Mathematik.)
Mainz, Germany
LOCALLY FINITE VERSUS FINITE SIMPLE GROUPS O. H. Kegel
Every countably infinite locally finite simple group
G
has a family
of finite
{S. } l
00
sUbgroups
with:
S. l
U
G
S. l
i=l
and there is a maximal normal subgroup Si+l
with
Si
simple group
n Ni + l = G
<1>.
may be viewed as a limit of the
is ordered by "involvement". easy observation: S./N. l
{S./N.} l
which
l
The following is an
If each of the finitt simple
is "known", then either
l
of
Thus, in a way, the
sequence of finite simple groups
groups
N + i l
G
contains
an infinite, elementary abelian p-subgroup for every prime
p , or there is a field
natural number a subgroup of
n
such that
GL(n, F)
G
F
and a
is isomorphic to
The essential point in 247
FINITE GROUPS
248
the proof of this is the fact that the finitely many two-parameter families of "known" finite simple groups have "nice" linear representation properties (and these may be given a form which allows us to define a "classically parametrized two-parameter family" of finite simple groups, for which the above theorem still holds) . Now, from the point of view of infinite groups, it seems rather unlikely that in a countable locally finite simple group the smallness with respect to one prime
p , i.e.
the fact that
elementary abelian p-subgroups are finite, should entail the smallness for all but one prime, and even for that prime
c , the characteristic of
the linearity of
imposes severe structural
G
restrictions on c-subgroups of
F,
G.
By the above theorem, the existence of a non-linear simple, locally finite group for
(at least) One prime
p
G
which
has only finite
elementary abelian p-subgroups (possibly
<1»,
entails the existence of a family of finite subgroups
{S.}
simple groups
~
as above, such that each of the {S./N.} ~
~
is "new".
Discussing locally finite groups in this
spirit, many questions about finite groups come up.
Here is one example, to which I would like
know the answer:
If
classical group
C(q)
GF(q)
~
GF(ql)
, then the
can in general be embeddec
in many ways into the classical group
of the automorphism group of
h~e
C(ql)
all of these embeddings equivalent under the
actic~
C(ql) ?
REFERENCE 1.
~~
O. H. Kegel, Uber Lokal endliche einfache Gruppen, Math. Zeit. 95 (1966).
AUTOMORPHIC ALGEBRAS Ernest Shult
A (not necessarily associative) algebra
A
is said to be automorphic if its automorphism group transitively permutes its one-dimensional subspaces. The object of this note is to report the following:
THEOREM 1.
Let
A
be a finite automorphic
algebra whose ground field
F
two elements.
A
Then either
contains more than
=
F
or
The following group-theoretic corollaries stem from this theorem:
COROLLARY 1.
Let
P
be a p-group of odd order
which admits a group of automorphisms transitively permuting its subgroups of order 251
p.
Then
P
is
252
FINITE GROUPS
abelian.
COROLLARY 2.
Suppose
G
is a group containing
one conjugate class of sUbgroups of odd prime order. Then a p-Sylow sUbgroup and only if
~l(S)
COROLLARY 3.
~
Let
S
of
G
is abelian if
Z(S)
p
be an odd prime.
Suppose
G
is a p-solvable group whose subgroups of order
p
are conjugate in the automorphism group of
Then
G
has p-length
1
G.
and abelian p-Sylow
subgroups. The 2-Sylow subgroups
S
of the Suzuki
groups are non-abelian 2-groups whose normalizer transitively permutes the involutions of this reason the requirement that
p
S
be odd is
necessary in Corollaries 1 through 3.
These groups
also give rise to an automorphic nil algebra over
GF(2)
A2 ~ O.
, for which
For
A
In addition,
examples exist in every dimension of automorphic non-nil algebras A2 ~ O.
A
over
GF(2)
for which
Thus Theorem 1 gives a best-possible
criterion on
dim (A)
and
that a finite
253 automorphic algebra be a zero-algebra. Corollary 3 is a generalization and sharpening of a theorem of Gaschutz and Yen [5] which concerned only the p-length conclusion. Corollary 1 was first considered by Eoen [1] and Higman [6].
Special cases of this corollary as
well as partial results on anticommutative automorphic algebras of odd characteristic have been obtained by a number of authors under special assumptions on subgroups of
dim A , Aut lA)
IFI
and transitive
[I, 2, 4, 6, 7, 8, 9, la].
The proof of Theorem 1 does not depend on these results, and its proof from first principles appears in [11] and [12].
The object of [11] was
the proof of the weaker
THEOREM 2. either
2 A
Under the hypothesis of Theorem I,
o
or
A
has no zero divisors.
The fact that Theorem 2 actually implies Theorem 1 is the content of [12].
Theorem 2, in
turn, is the consequence of the following more general result, which seems to isolate the essential mechanics of what is going on:
254
FINITE GROUPS
THEOREM 3.
Let
algebra over
GF(q)
in a
A
A
be a finite dimensional and let
A
La : B
~
induces a nilpotent linear transformation B.
Suppose
automorphisms
G
A
admits a group of
which transitively permutes the
one-dimensional subspaces of
=
be a left ideal
Suppose left multiplication by any element
of
AB
B
B.
If
q > 2 , then
0 . Perhaps a few sketchy remarks concerning the
proofs of these theorems would be appropriate. The version of Theorem 3 appearing in [11] is augmented by the additional hypothesis that B
2
= O.
This alone implies Theorem 2 and hence
the fact that the above satisfies follows: over A
Let
GF(q)
B A
involved in Theorem 3
"B"
2
=
0
This can be seen as
be a finite automorphic algebra
q > 2 , and let
as a G-module and let
module isomorphisms.
W l
G = Aut A and
W 2
The G-module
A
View
be two
=
Wl(A) ffi W2 (A)
can be converted into an algebra by defining (Wl(A»
2
=
Wl(A) 'W 2 (A)
(W 2 (A»
2
W2(A) 'Wl(A)
by the rule
wl(a)
By a theorem of Chevalley [3J
=
0
and products
.w 2 (b) =
w2 (ab)
if an automorphic
255
algebra
A
has zero divisors, it is a nil-algebra
(left multiplications are nilpotent transformations). Now
G
acts on
A
in such a manner that
play the role of
].12 (A)
A
and
in the augmented Theorem 3. 2
implies
=
].12 (A )
0
or
B
0
=
,
and
respectively
Then
A2
A
A].12(A)
= 0
since
is a
module isomorphism. The proof of Theorem 3 utilizes induction on
= nand
dim A,
dim B
stages.
The first is to "shape up" the algebra so
that
A
W ffi B
and both
Wand
w2
where B
IGI
=
' and occurs in three
B2
=
BW
0,
=
WB
=
B
are irreducible G-modules.
The second stage is a reduction to the case that p
=
char F
From this
does not divide
point on, the nilpotence of the left-multiplications produces a congruence
where G
q
=
IFI
and
C
acting irreducibly on
is a cyclic subgroup of B.
theoretic case analysis forces G '" LF(2, n + 1)
or
A largely numberdim B
<
14 , and
LF(2, 2n + 1)
In proving Theorem 1 from Theorem 2, it suffices to show that for a finite automorphic
256
FINITE GROUPS
algebra
A
over
GF(q)
, the following hypotheses
are inconsistent: (i) (ii) (iii)
A
2
~ 0
q > 2 A
~
GF(q)
An extension of Chevalley's methods yields
LEMMA.
Let
F
be a finite field and let
be
R
the vector space of n-tuples with entries from Let in
p
be a homogeneous polynomial of degree
F[Xl, ... ,xnJ
Suppose
linear transformations of
G R
F
n
is a group of
which acts transitive-
lyon the one-dimensional subspaces of
Rand
suppose p(v) for every (i) or
(ii)
v E Rand p(v)
for some
for all
R
G
of a one-dimension-
fixes every vector in that
i.e. for any
AE F
Then either v E R
the stabilizer in
al subspace of subspace:
0
g E G.
implies
v E R g E G,
A
=
1
or
v
=
vg 0
AV
257
With the aid of this lemma and Theorem 2, hypotheses (i)-(iii) (i v)
imply
IAut
(g - 1,
(A)
I)
1 .
This result assumes the role that the hypothesis of nilpotent multiplication played in the proof of Theorem 3:
To provide enough number-theoretic
ammunition to dispose of the case that a g'-group of automorphisms can transitively permute the one-dimensional subspaces of
A.
Thus
has a non-trivial p-Sylow subgroup p = char GF (g) neither does of A
P
in
A
Since A
'
1
A
Aut (A)
P , where
has no zero-divisors,
the sub-algebra of fixed points
It follows that (iii)
fails for
' and so everyone-dimensional subspace of
l
is a sub-algebra.
In this case
A
contains two
linearly independent idempotent elements, y , and that for any (v)
If g
A
=
(x
+ 8y)
A
x
and
8 E GF(g)
2
2
x + 8 Y + 8(xy + yx)
were commutative,
2 , against (ii).
We may now define a new
8
Thus
= 0 A
algebr~
or
1 , whence
is not commutative. B
=
A(+,
0)
where
258
B
FINITE GROUPS
=
A
as vector spaces over
product
11
x Then
"
and a new
is defined by xy + yx .
y
0
B
o
GF(q)
admits
Aut
(A)
and is a commutative
automorphic algebra satisfying (ii)-(v). follows that Since 2
A
and
B
2
=
0
so
A
is anti-commutative.
has no zero divisors
A
It
q
is a power of
is commutative, a contradiction.
REFERENCES 1.
J. Boen, On p-automorphic p-groups, Pac. J. Math. vol. 12
2.
(1962),
813-816.
J. Boen, O. Rothaus, and J. Thompson, Further results on p-automorphic p-groups, Pac. J. Ma th. vo 1. 12 ( 1962), 817- 8 21 .
3.
C. Chevalley, Demonstration d'une hypothese de M. Artin, Abh. Math. Sem. U. Hamburg vol. II
4.
(1936).
L. Dornhoff, p-automorphic p-groups and homogeneous algebras, preprint, Yale University.
259 5.
W. Gaschutz and T. Yen, Groups with an automorphism group which is transitive on the elements of prime order, Math. Z. 86 (1964), 123-127.
6.
G. Higman, Suzuki 2-groups, Ill. J. Math. 7 (1963),73-96.
7.
A. I. Kostrikin, On homogeneous algebras, Izvestia Acad. Nauk SSSR, 29 (1965), 471-483.
8.
D. Passman, personal communication, fall 1967.
9.
E. Shult, On semi-p-automorphic groups, I. Submitted to Pac. J. Math (February, 1966). , On semi-p-automorphic groups, 11.
10.
Submitted to Pac. J. Math (April, 1966). , On finite automorphic algebras,
11.
Submitted to Ill. J. Math. 12.
(February 1968)
, On the triviality of finite automorphic algebras, submitted to Ill. J. Math.
(April 1968).
AUTOMORPHISMS OF FINITE GROUPS Chih-Han Sah
A rather general question in group theory is: Given a finite group
G , what is
the relation between the normal structure of
G
and the normal
structure of the automorphism group Aut (G)
?
Schreier conjectured that is solvable when
G
Aut (G)/Inn (G)
is a finite simple group.
During the 1963 Harvard Symposium, Brauer verified the Schreier conjecture when the 2-Sylow subgroup of
G
has certain special forms.
generalized this result.
Later, Glauberman
The difficulty in
applying Glauberman's result is that we need to know something about the 2-Sylow subgroup or its 261
262
FINITE GROUPS
embedding in
G.
For all the known simple groups,
the Schreier conjecture has been verified. It appears reasonable to insert a number of groups between
Aut (G)
and
Inn (G)
in such a
way that they reflect the normal structure of
G
From the point of view of representation theory, two candidates appeared in Burnside's work: A (G) c
=
group of all automorphisms fixing all complex characters. group of all automorphisms carrying each complex character onto an algebraic conjugate (over
~)
.
It is clear that other such groups can be introduced. It is easy to see that we have a normal series:
Using Brauer's permutation lemma, one sees that
Af(G)
is the group of all automorphisms
preserving the conjugate classes of cyclic subgroups of
G
and
A (G) c
is the group of
automorphisms preserving the conjugate classes of G.
It is then easy to show that
Af(G)/Ac(G)
is
263
an abelian group. Schreier's conjecture can obviously be split up into two halves. Burnside claimed that, for any finite group G ,
A (G)/Inn (G) c
is always abelian.
At first,
it was not certain if this group could be nontrivial.
Later Burnside, G. E. Wall and others
showed that
A (G)/Inn (G) c
abelian groups.
could be non-trivial
In a forthcoming paper in the
Journal of Algebra we constructed p-groups (for any prime
p) for which
non-abelian.
is A (G)/Inn (G) c Using cohomological arguments, we
can show:
THEOREM.
If all the composition factors of the
finite group then
G
satisfy Schreier's conjecture,
Ac(G)/Inn (G)
is solvable.
It is known to Burnside that the primes dividing
lAC (G) I
must divide
IGI
.
Nothing much
is known about the general structure of
Harvard University and University of Pennsylvania
