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G, 9 f---+ g-l and a binary operation G 2 -> G, (g, h) f---+ gh. A universal sentence of La is one of the form 'v'x
en
134
tuple of distinct variables and w(x) is a word in at most the variables in x. Henceforth we shall omit the universal quantifiers and abbreviate the law I;fx(w(x) = 1) as simply w(x) = 1. From this point on we tacitly assume the group axioms. The trivial variety E, determined by the law x = 1, is the isomorphism class of the one element group. All other varieties of groups are nontrivial. Every nontrivial variety V of groups admits, for each cardinal r, groups free of rank r relative to V. We adopt the notation Fr(V) for a fixed, but arbitrary, group free of rank r relative to V. Any two such groups (for fixed V and r) are isomorphic. Convention: If E is the trivial variety and r is any cardinal, then Fr(E) is the trivial group 1. If G and H are groups then we say that G universally covers H provided every universal sentence of Lo true in G is also true in H. Note that H :::; G is a sufficient condition for G to universally cover H. We say that G and H are universally equivalent or have the same universal theory and write G ="1 H provided G universally covers Hand H universally covers G. An existential sentence of Lo is one of the form :lXip(x) where x is a tuple of distinct variables and ip(x) is a formula of Lo containing no quantifiers and containing free at most the variables in x. A primitive sentence of Lo is an existential sentence of Lo equivalent modulo the group axioms to one of the form :lx(l\j(Uj(x) = 1) 1\ I\k(Vk(X) -=I- 1)) where x is a tuple of distinct variables and the Uj(x) and Vk(X) are words in at most the variables in x. Since the negation of a universal sentence of Lo is equivalent to an existential sentence of Lo and vice-versa G ="1 H may be paraphrased as asserting that every universal sentence and every existential sentence of Lo true in G is also true in H (and vice-versa). It is easy to see that if G 1 ="1 G ="1 G 2 , then G 1 :::; H :::; G 2 is a sufficient condition for G ="1 H. A somewhat more sophisticated sufficient condition is the following. H :::; G and every finite system
Uj(X)
=
1,1:::;j:::; J
Vk(X) -=I- 1,1 :::; k :::; K of equations and inequations ( in finitely many variables x = (x 1, ... , X n ) ) which has a solution in G must also have a solution in H. To see that this is so observe that G universally covers Hi so, it will suffice to show that every existential sentence of Lo true in G must also be true in H. Now we may assume that the matrix ip(x) of the existential sentence :lXip(x) is written in disjunctive normal form modulo the group axioms Vi(l\j(Ui,j(X) = 1) 1\ I\k(Vi,k(X) -=I- 1)). The sentence is then equivalent to
135
the disjunction VaX(!\j(Ui,j(X) = 1) !\ !\k(Vi,k(X) =I- 1)) of primitive sentences. Since a disjunction is true provided at least one of the disjuncts is, it suffices to prove that every primitive sentence of Lo true in G is also true in H; hence, the sufficiency of the above criterion is established. We say that two groups G and H are elementarily equivalent and write G = H provided G and H satisfy the same sentences of Lo. A theorem of Vaught (Theorem 4 of Chapter 6, Section 38 in [G]) asserts that, if V is any variety and rand s are infinite cardinals, then Fr(V) Fs(V). In particular, Fr(V) ='<1 Fs(V) when rand s are infinite. Let n be a positive integer. The Burnside variety Bn of exponent n is the variety of groups determined by the law xn = 1. Adian proved that, for all sufficiently large odd n, (B1) Fr(Bn) is infinite for all r ~ 2; moreover, every finite subgroup of Fr(Bn) is cyclic. and (B2) The centralizer of every nontrivial element in Fr(Bn) is cyclic for all r ~ 2. Sirvanjan proved that, for all sufficiently large odd n, (B3) Fw(Bn) embeds in F2(Bn). We shall call an integer n > 0 an Adian-Sirvanjan integer provided (B1), (B2) and (B3) hold for Bn. A prime Adian-Sirvanjan integer p shall be an Adian-Sirvanjan prime. Every member G of a variety V of groups is a homomorphic image of a group Fr(V) free in V. If there is an epimorphism 'ljJ : Fr(V) ~ G such that r is finite and K ere 'ljJ) is the normal closure in Fr (V) of finitely many elements of Fr(V), then G is finitely presented relative to V.
=
3. Varieties and Discrimination We begin by remarking that, although we chose to live in the world of groups, the results of this section go through in the context of universal algebra. Let V be a nontrivial variety of groups. Let Fw (V) be a group free of count ably infinite rank relative to V. Suppose {aI, a2, ... } = {an+l : n < w} freely generates Fw(V) relative to V. Definition 3.1. [NJ A group G E V discriminates V provided G discriminates Fw(V).
136
Now G E V discriminates V just in case, given finitely many elements wk(al, ... , an)
=1=
1
in Fw(V), there is a homomorphism 'ljJ : Fw(V) --+ G such that 'ljJ( Wk( aI, ... , an)) =1= 1 for all k. This is equivalent to the following. Given finitely many words Wk(Xl, ... , xn) such that none of the equations Wk(XI, ... ,Xn )
=1
is a law in V, there is a tuple (g1, ... , gn) E Gn such that Wk(gl, ... , gn) for all k.
=1=
1
Convention: The trivial group 1 discriminates the trivial variety E. Definition 3.2. Let V be a variety of groups. V is finitely discriminable provided there is a finitely generated group G E V such that G discriminates V. Lemma 3.1. V is finitely discriminable if and only if there is a positive integer r such that Fr (V) discriminates V. Proof: If G = Fr(V) discriminates V for some integer r > 0, then V is discriminated by an r-generator member; hence, it is finitely discriminable. Suppose V is finitely discriminable. Suppose r is a positive integer and G = (b l , ... , br ) E V discriminates V. Let Fr(V) be freely generated relative to V by aI, ... , ar· Then we get an epimorphism 'ljJ : Fr(V) --+ G, ai 1-+ bi, 1 :::; i :::; r. Suppose that Wk(Xl, ... , xn) are finitely many words such that none of the equations Wk(XI, ... , xn) = 1 is a law in V. Then there are elements gi = ui(b1, ... , br ), 1 :::; i :::; n in G such that Wk(UI(b l , ... , br ), ... , un(b 1 , ... , br )) =1= 1 for all k. It follows that Wk( UI (aI, ... , a r ), ... , Un (aI, ... , a r )) =1= 1 in Fr(V) for all k. Hence, Fr(V) discriminates V . •
Definition 3.3. Let V be a finitely discriminable variety of groups. Then min{l :::; r < w : Fr(V) discriminates V} is the index of discrimination of V. If m is the index of discrimination of V, then D(V) = {Fr(V) : m :::; r < w}. Theorem 3.1. fGS} Let V be a finitely discriminable variety of groups. Let r ;::: 1 be a cardinal. Then Fr(V) ='1 Fs(V) for all cardinals s ;::: r if and only if Fr (V) discriminates V. In particular, if m is the index of discrimination of V, then Fm(V) ='1 Fs(V) for all m:::; s:::; w.
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Theorem 3.2. Let V be a finitely discriminable variety of groups with index of discrimination m. Let G E V be Fm(V) inclusive. If D(V) discriminates G, then G ='1 Fm(V). Proof: Assume D(V) discriminates G. It will suffice to show that, if Uj(XI, ... ,X n )
=
1
Vk(XI, ... , Xn)
-I-
1
has a solution (gl, ... ,gn) E Gn, then it has a solution over Fm(V). Since D(V) discriminates G there is an integer r ~ m and a homomorphism 'lj; : G ----) Fr(V) such that 'lj;(Vk(gl, ... ,gn)) -I- 1 for all 1 ::; k ::; K. It follows that the primitive sentence 3XI, ... ,xn(Aj(uj(XI, ... ,Xn) = 1) A Ad Vk (Xl, ... , Xn) -I- 1)) is true in Fr (V). But since Fr (V) ='1 Fm (V) the above primitive sentence must also be true in Fm (V). Hence, the system has a solution over F m (V) . •
Corollary 3.1. Let V be a finitely discriminable variety of groups with index of discrimination m. Let G E V be finitely presented relative to V and suppose that G is Fm(V) inclusive. Then G ='1 Fm(V) if and only if D(V) discriminates G. Proof: One direction follows immediately from the theorem. Suppose G E V is finitely presented relative to V, is Fm(V) inclusive and G ='1 Fm(V), Suppose < al, ... ,an;RI, ... ,RJ >vis a finite presentation of G relative to V. Let wk(al, ... , an), 1 ::; k ::; K be finitely many nontrivial elements of G. Then the primitive sentence 3XI, ... ,xn(Aj(Rj(XI, ... ,Xn) = 1) A Ak (Wk (Xl, ... , Xn) -I- 1)) holds in G; hence, it holds in Fm (V) and there is (b l , ... , bn ) E Fm(v)n such that
Rj(b l
, ... ,
bn ) = 1
wk(h, ... , bn )
-I-
1
1::; j ::; J 1::; k ::; K.
It follows that the assignment ai I---> bi , 1 ::; i ::; n extends to a homomorphism'lj; : G ----) Fm(V) such that 'lj;(wk(al, ... , an)) -I- 1, 1 ::; k ::; K.
•
4. The Variety 0 of All Groups In this section we merely review known results about universally free groups (see Definition 4.1). These results will be contrasted later with results for the groups G ='1 F2(Bp) where p is an Adian-Sirvanjan prime. If 0 is the variety of all groups and r ~ 1 is a cardinal, then we write Fr for
138
Fr(O). Fw embeds in F2. For example, the commutator subgroup [F2' F 2] of F2 is free of countably infinite rank. Now let 2 S; r S; w. Then Fw ~ [F2' F2] S; Fr S; Fw from which it follows that Fr ='1 Fs for all cardinals 2 S; r < s. Thus 2 is the index of discrimination of O. The universal equivalence of the nonabelian free groups suggests the possibility of their elementary equivalence. Of course their universal equivalence is a far cry from their elementary equivalence. Suppose R is a commutative ring with 1. Within the category of unital R-modules an object P is projective just in case every short exact sequence O-+N-+M-+P-+O
splits. This is easily seen to be equivalent to P being a direct summand in a free R-module. Now suppose V is a variety of groups. We define a group P E V to be projective relative to V provided every short exact sequence of groups in V, 1 -+ K
-+
G
-+
P
-+
1,
splits. Essentially the same proof shows that this is equivalent to P being a retract of a group free in V. By the Neilsen-Schreier subgroup theorem, a group P is projective relative to the variety of all groups if and only if it is free. The Neilsen-Schreier subgroup theorem also implies that a group is freely separated (discriminated) if and only if it is residually (fully residually) free. Suppose G is a nonabelian residually free group. Suppose gh =f=. hg in G. Then their commutator [g, h] = g-lh-1gh is nontrivial. Thus, there is a free group Fr and an epimorphism 'ljJ : G -+ Fr such that ['ljJ(g) , 'ljJ(h)] = 'ljJ([g, h]) =f=. 1. Hence, Fr is nonabelian and r ~ 2. Since Fr is projective it is a retract in G and F2 S; Fr S; G; so, G is F2inclusive. Nonabelian free groups, of course, satisfy the existential sentence ::Ix, y( xy =f=. yx). Other properties of nonabelian free groups are that they are CT and even CSA. Here a group is GT or commutative transitive provided the centralizers of nontrivial elements coincide with the maximal abelian subgroups. That is rendered by the universal sentence
\:Ix, y, z(((y =f=. 1) 1\ (xy
= yx) 1\ (yz = zy))
-+
(xz
= zx)).
Moreover, a group is GSA or conjugately separated abelian provided maximal abelian subgroups are malnormal. That is equivalent to being CT and satisfying the universal sentence
\:Ix,y,z(((x =f=.1) 1\ (xy Free groups are CSA.
= yx) 1\ (Z-lyzx = xz-1yz))
-+
(xz = zx)).
139
Lemma 4.1. fE] A GT residually free group is GSA.
Proo!' Suppose G is eT and residually free. Let a E G\ {I} and suppose b,g-lbg E Gc(a) = {x E G : ax = xa}. Suppose to deduce a contradiction that 9 rf- Gc(a). Then [g,a] =I=- 1. Thus there is a free group F and an epimorphism 'l/J : G ----+ F such that ['l/J(g), 'l/J(a)] = 'l/J([g, a]) =I=- 1. But this is impossible. From ['l/J(g), 'l/J(a)] =I=- 1 we have 'l/J(a) =I=- 1. Moreover 'l/J(b), 'l/J(g)-l'l/J(b)'l/J(g) E GF('l/J(a)) and F is eSA. Hence, 'l/J(g) commutes with 'l/J(a) - a contradiction. Hence g E Gc(a) and Gis eSA . • fE] Let G be a nonabelian residually free group. The following three conditions are equivalent in pairs.
Theorem 4.1.
(1) G is fully residually free. (2) G is GT. (3) G is GSA.
Proo!' Lemma 4.1 has already established the equivalence of (2) and (3). It will suffice to show that (1) and (2) are equivalent. Suppose G is fully residually free. Let b E G\{l} and suppose a, c E Gc(b). Assume to deduce a contradiction that ac =I=- ca Then [a, c] =I=- 1. Thus there is a free group F and an epimorphism 'l/J : G ----+ F such that 'l/J(b) =I=- 1 and ['l/J(a), 'l/J(c)] = 'l/J([a, cD =I=- 1. But this is impossible. 'l/J(a)'l/J(b) = 'l/J(b)'l/J(a), 'l/J(b)'l/J(c) = 'l/J(c)'l/J(b) and F is CT; hence, 'l/J(a)'l/J(c) = 'l/J(c)'l/J(a) - a contradiction. Therefore ac = ca and G is eT. Now suppose G is CT. The proof will proceed by induction on the cardinality n of S = {gl, ... , gn} <;;; G\ {I} no element of which is annihilated in a free homomorphic image. The result is true when n = 1 since G is residually free. Now suppose n > 1 and the result is true for all 1 :::; k < n. Suppose first that S is not contained in an abelian subgroup of G. Then some pair of elements of S, which we may take to be gn-l and gn, does not commute. Thus T = {gl, ... ,gn-2,[gn-l,gn]} is contained in G\{l} and, by inductive hypothesis, there is a free group F and an epimorphism 'l/J : G ----+ F such that 'l/J does not annihilate any element of T. But then 'l/J cannot annihilate any element of S either. It remains to treat the case where the gi commute in pairs ,which hypothesis we now assume. Since G is a residually free CT group it is eSA by Lemma 4.1. We claim that there is some 9 E G such that g-lgng does not commute with gn-l. Otherwise, since G is eSA, gn-l would be central in G. But a nonabelian eT group must be centerless; so, we have arrived at a contradiction. The claim is
140
established. Pick one such g. Hence U = {gl, ... ,gn-2,[gn-1,g-l gng ]} is contained in G\{l}. By inductive hypothesis there is a free group F and an epimorphism 'ljJ : G -4 F such that 'ljJ does not annihilate any element of U. But then 'ljJ cannot annihilate any element of S either. That completes the induction . •
Definition 4.1. A group G
='<1
F2 is universally free.
(R) A finitely generated group is universally free if and
Theorem 4.2.
only if it is nonabelian and fully residually free.
(KM1) A finitely generated fully residually free group is
Theorem 4.3. finitely presented.
Definition 4.2. Let G be a CT group and let a E G\{l}. Let M Then the HNN-extension (G,
t; rel(G), C 1 mt = m \1m
E
= Cc(a).
M)
is a free rank 1 centralizer extension of G.
The above construction preserves fully residually freeness. In particular, we have
Theorem 4.4. If Go is fully residually free then so is every free rank 1 centralizer extension of Go. Fro of: By Definition 4.2, we take b E G o\{l} and let M = Cco(b). So that our free rank 1 centralizer extension of Go is the group G, G
= (Go, t;
rel(G o ), C 1 zt
= z \lz EM).
Start off by viewing G as the amalgamated free product G
= Go *M (M x (t; ).
We need to show that G is fully residually free. For that purpose let gl, ... , gk be finitely many nontrivial elements of G. Using the normal form for free products with amalgamation (see [MKSj), we may write for each j = 1, ... , k
141
where N(j) :?: 0, bi,j E Go \M, mi,j E iZ\{O}, and Zj E M. Note that bi,j E Go \M is equivalent to [bi,j, b] =f. 1. Now since Go is fully residually free, there is a free group F and an epimorphism 'P : Go ----) F such that ['P(bi,j),'P(b)] = 'P([bi,j,b]) =f. 1 for all i,j. This forces 'P(bi,j) =f. 1 and 'P(b) =f. 1. Let GF('P(b)) = (u). Suppose that 'P(Zj) = u ej for all j, 1:::; j :::; k. Now for each positive integer n E N, we may define an extension 'ljJn : G ----) F of 'P by 'ljJn \c o = 'P, 'ljJn(t) = un. Now fix a j, 1 :::; j :::; k. Could we have 'ljJn(gj) = 1 for infinitely many n E N? Suppose to deduce a contradiction that there were infinitely many n E N such that 'ljJn(gj) = 1. Then 'P(bo,j )uml,jn'P(b1,j )"''P(b N (j)-l,j )umN(i),jn+e j
=1
for infinitely many values of n. But then by G. Baumslag's "Big Powers Lemma" (see Proposition 1 [GB]), we then conclude that
for some i, with 0 :::; i :::; N(j) - 1. Thus for that 'P(bi,j) we must have that 'P(bi,j) E GF(u) = GF('P(b)) and so ['P(bi,j),'P(b)] = 1. This contradicts our choice of 'P. The above contradiction shows that the set
is a cofinite subset of N (Le., its complement S; = N\Sj is finite). Since this is so for all j, 1 :::; j :::; k, we must have the finite intersection Sl n ... n Sk
=f. ¢.
(Note if Sl n··· n Sk were empty, then (Sl n··· n Sk)' = S~ u··· u S~ = N - which is impossible since S~ U ... U S~ is a finite union of finite sets.) Choose n E Sl n ... n Sk. Then 'ljJn(gj) =f. 1 for all j with 1 :::; j :::; k. Hence G is fully residually free . • Theorem 4.5. [KM1] A finitely generated group G is fully residually free if and only if there is a finite rank free group Go and a finite sequence Go :::; G 1 :::; ... :::; G n of free rank 1 centralizer extensions such that G is isomorphic to a finitely generated subgroup of G n .
142
5. The Burnside Varieties Let n be an Adian-Sirvanjan integer. Then Fw(Bn) embeds in F2(Bn). So, if 2 ::; r ::; w, then F2(B n ) embeds in Fr(Bn) which, in turn, embeds in F2(Bn). It follows that Fr(Bn) =\;/ F2(B n ) for all 2 ::; r ::; w. Thus 2 is the index of discrimination of Bn. Moreover, since the centralizer of every nontrivial element in Fr(Bn) is cyclic, the relatively free groups Fr(Bn) are CT. Now suppose that p is an Adian-Sirvanjan prime.
Lemma 5.1. Suppose G E Bp and G is Gp x Gp exclusive. If G is GT, then G is GSA.
Proof: Suppose G E B p, G is GpxGp exclusive and G is CT. Let a E G\{l}. Then Gc(a) = (a) ~ Gp. Suppose g-Ibg E (a) for some 1 i- b E (a). Since (a) = (b) we may assume b = a. Then g-Iag = am and a = g-Pag P = amP. But we may compute exponents modulo p and, by Fermat's Little Theorem,
amP = am. Therefore, am = a and g-Iag = a. So 9 E Gc(a) = (a). Hence, Gis CSA . •
Corollary 5.1. The relatively free groups Fr(Bp) are GSA.
Proof: Since the centralizer of every nontrivial element is isomorphic to Gp, one has that Fr(Bp) is CT and Gp x Gp exclusive . • More generally, if n is any Adian-Sirvanjan integer, then the free groups Fr(Bn) are CSA. To see that let G = Fr(Bn) where r 2': 2. Let a E G\ {I}. Let Gc(a) = (b). Suppose g-I(b)g intersects (b) nontrivially. Since G is CT we must have, in that event, g-I(b)g = (b). Consider the subgroup H = (g, b) ::; G. Observe (b) is normal in H and the quotient H / (b) is generated by the image of 9 - an element of finite order. Then IHI = I(b) I[H : (b) 1< 00. Then H must be cyclic. But Gc(a) = (b) ::; H is maximal cyclic. Hence, H = (b) and 9 E (b) = Gc(a). It follows that Gc(a) is malnormal in G. Hence, G is CSA. Theorem 5.1. Suppose G E Bp is GT and Gp x Gp exclusive. If G is separated by D(Bp), then G is discriminated by D(Bp).
Proof: The proof will be by induction on the cardinality n of S = {gl, ... ,gn} <;;; G\{l} no element of which is annihilated by a homomorphism into a group free in Bp' We have the result for n = 1 since G is separated by D(B p ). Now suppose n > 1 and the result is true for all k with 1 ::; k < n. Assume first that S is contained in an abelian subgroup
143
of G. Then (gl) = ... = (gn) ~ Cp and so gi = g'{'i where p does not divide mi for 2 ::; i ::; n. Now, since D(Bp) separates G, there is r ::::: 2 and a homomorphism '¢ : G -; Fr(Bp) such that ,¢(gl) # 1. But then ,¢(gi) = ,¢(gl )mi # 1 for all 2 ::; i ::; n. Hence, '¢ does not annihilate any element of S. Now suppose at least one pair of elements of S does not commute. We may assume that gn-1 and gn do not commute. Then T = {gl, ···,gn-2, [gn-1,gn]} is contained in G\{l} By inductive hypothesis there is r ::::: 2 and a homomorphism '¢ : G -; Fr (Bp) such that '¢ does annihilate any element of T. But then '¢ does not annihilate any element of S either. That completes the induction . • Observe that, since F2(Bp) satisfies the universal sentence I::/x, y(((x "lI) 1\ (xy = yx)) -; V~:~(y = xk)), every group G ='1 F2(Bp) is Cp x Cp exclusive. More generally, if n is any Adian-Sirvanjan integer and G ='1 F 2 (B n ), then, for every 9 E G\{l}, one has that Ca(g) is cyclic. To see that consider the universal sentences (ul) I::/X1,X2((X1X2 = X2X1) -; VO::;k 1 ,k 2,ml,m2
= XjXi) -; Vi<j(Xi = Xj)).
Both hold in F 2 (B n ) since every abelian subgroup of F 2 (B n ) is cyclic of order at most n. Hence they both hold in G. (ul) asserts that every abelian subgroup is locally cyclic and (u2) asserts that every abelian subgroup has at most n elements. Suppose that n is a composite Adian-Sirvanjan integer. Let 1 < d < n be a divisor of n. Let B be freely generated in Bn by {a1, a2, a3} and let A be the subgroup generated (necessarily freely) by {a1, a2}. Let G be the subgroup of B generated by {a1,a2,al}. SinceF2 (B n ) = A ::; G ::; B = F3(Bn) ='1 F2(B n ) we have G ='1 F2(Bn). But the finitely generated group G cannot be free in Bn since its abelianization, isomorphic to Cn x Cn X Cd, has order n 2 d which is not a power of n. For an Adian-Sirvanjan prime p are there any finitely generated G ='1 F2 (Bp) which are not free in Bp? We ponder that question in the next section. We conclude this section with a proof that cyclicity of finite subgroups is inherited by models of the universal and existential theory of the free Burnside groups.
Theorem 5.2. Let n be an Adian-Sirvanjan integer and assume G F2(Bn). Then every finite subgroup of G is cyclic.
='1
144
Proof' For each positive ineger N the universal sentence \Ix 1 , ... , xN(Ai$.j Vk (XiXj
=
Xk) -; Ai<j(XiXj
=
XjXi))
holds in F2(Bn) since every finite subgroup is abelian. Thus every finite subgroup of G is abelian. But we have already seen that every abelian subgroup of such G is cyclic . •
6.
A Possible Non-Free Model and a Question of Philip Hall
If G is a group and H ::; G let HG be the normal closure of H in G. If H, K ::; G let [H, K] be the subgroup generated by {[h, k] : (h, k) E Hx K}. If G 1 and G 2 are groups let G 1 * G 2 be their free product. Let V be a variety of groups and G be a group. Let V(G) be the intersection of the family of subgroups K normal in G such that GIK E V. Then V(G), the verbal subgroup of G corresponding to V, is fully invariant in G and is the least normal subgroup K in G such that G IKE V. We define (following Hanna Neumann [N] and Magnus, Karrass, Solitar [MKS]) the verbal product G 1 *v G 2 as r/([G 1, G 2]r n Vcr)) where r = G 1 * G 2 . If G 1, G2 E V then G 1 *v G 2 E V and each of G 1 and G 2 embeds in G 1 *VG2' Moreover *v restricted to groups in V is the coproduct in V. That means essentially that if G 1 , G 2, G E V then every pair of homomorphisms 1/;i : G i - ; G, i = 1,2, uniquely determines a homomorphism 1/; : G 1 *VG2 -;
G. Among other results one has that Fr(V) is the verbal product relative to V of r copies of F 1 (V). Moreover, if G 1 , G 2 E V, then each of G 1 and G 2 is a retract of G 1 *v G 2. We observe G 1 is a retract since if 1/; : G 1 *v G 2 -; G 1 is the homomorphism determined by 1/;1 : G 1 -; G 1 and 1/;2 : G 2 -; G 1 where 1/;1 is the identity automorphism and 1/;2 is the trivial map 1/;2 (x) = 1 for all x, then 1/; is a retraction from G 1 *v G 2 onto G 1 . Similarly G 2 is a retract of G 1 *v G 2. Furthermore, G 1 *v G 2 is generated by the embedded images of G 1 and G 2 and, if Hi ::; G i , i = 1,2, then the subgroup (H1' H 2) of G 1 *v G 2 has the verbal product decomposition H1 *v H 2. Now let p be an Adian-Sirvanjan prime. Let *pdenote the verbal product with respect to the variety Bp and let Kp be the Kostrikin variety of locally finite groups satisfying the law x P = 1. Let Kp be the verbal subgroup operator corresponding to Kp- Suppose r ~ 2 is finite. Then Fr(Kp) = Fr(Bp)/Kp(Fr(Bp)) is a finite group. Since Kp(Fr(Bp)) has finite index in the finitely generated group Fr(Bp) it must itself be finitely
145
generated. However, I\:p(Fr(Bp)) is perfect (i.e. it coincides with its commutator subgroup). Therefore it cannot be free in Bp. This is so since the abelianization of Fd(Bp) for any cardinal d ?: 1 is a vector space of dimension d over the p element field. Now let F = F4(Bp) be freely generated relative to Bp by a1, a2, a3 and a4. Let A be the subgroup generated (necessarily freely) bya3 and a4 and let B be the subgroup generated (necessarily freely) by a1 and a2. Let C = I\:p(A) so that C is finitely generated and perfect. F = B *p A. Consider the subgroup 0 = (B, C) = B *p C. Now F2(Bp) = B :::; 0 :::; F = F4(Bp) ='V F2(Bp). It follows that the finitely generated group 0 is universally equivalent to F2(Bp). Observe that, if 0 were free in B p, then the retract C of 0 would be projective relative to Bp. Here we observe a connection with Problem 21 of [NJ, which problem is attributed to Philip Hall. The question posed is the following. Suppose V is a variety of groups of exponent zero or prime power. If P E V is projective relative to V must P be free in V? Note that a positive answer to Philip Hall's question in the case of exponent an Adian-Sirvanjan prime would imply that 0 is not free in Bp. Kovacs and Newman [KN] report that Philip Hall's question has a negative answer in the case of exponent zero; however, to the best of our knowledge the question remains open for prime power exponent. Other conditions would also imply that 0 is not free in Bp. Suppose we define the Rank of a group to be the minimum cardinality of a set of generators. A consequence of the Grushko-Neumann Theorem asserts that Rank(Ol * O 2) = Rank(Ol) + Rank(02)' Now C = C' :::; 0' so 0/0' ~ Cp x Cpo If 0 were free it would have rank 2 and, since it is nonabelian, Rank( 0) = 2 also under the assumption of freeness. But, if it were the case that Rank(Ol *p O 2) = Rank(Ol) + Rank(02) for all 0 1, O 2 E B p, we would have Rank(O) = Rank(B) + Rank(C) = 2 + Rank(C) and Rank( C) > 0 since C -=J- 1. Thus, if the analog of the Grushko-Neumann corollary holds for B p, then 0 cannot be free in Bp. Suppose the finite rank free groups Fr(Bp), 2:::; r < ware Hopfian. As just argued, if 0 were free in B p , then rank(O) = 2. Say 0 were freely generated by bland b2. Now let 'Ij; : 0 -+ 0 be the endomorphism determined by 'lj;1 : B -+ 0, 'lj;2 : C -+ 0 where 'lj;1 is the homomorphism determined by ai f---4 bi , i = 1,2, and 'lj;2 is the trivial map 'lj;2(X) = 1 for all x. Then 'Ij; is an epi-endomorphism with 1 -=J- C :::; Ker('Ij;). That would contradict the Hopf property. Hence, if the free groups Fr(Bp), 2 :::; r < ware Hopfian, then 0 cannot be free in Bp. As far as we know it also is an open question as to whether or not these free groups Fr(Bp) are Hopfian.
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7. Questions Let G be a group and let n be a positive integer, Let (Xl, .. " Xn; ) be free on the n distinct elements Xl, .. " Xn , Let wE G* (Xl, .. " Xn; ), View the formal expression w = 1 as an equation over G in the variables Xl, .. " X n , Now every assignment Xi I---? gi E G, i = 1, .. " n, extends to a unique retraction 'ljJ : G * (Xl, .. " Xn; ) .-, G, Call the tuple (gl, .. " gn) E Gn a solution to w = 1 provided w E Ker('ljJ) , For each subset S ~ G * (Xl, .. " Xn; ), let Vc(S) ~ Gn be the solution set to the system w = 1, w E S of equations, G is equationally Noetherian provided for every positive integer n and every subset S ~ G * (Xl, .. " Xn; ) there is a finite subset So ~ S such that Vc(S) = Vc(So), Question 1: If n is an Adian-Sirvanjan integer and r 2: 2 is an integer must Fr(Bn) be equationally Noetherian? Question 2(Philip Hall): If V is a variety of groups of prime power exponent and P E V is projective relative to V must P be free in V? Question 3: Suppose we define the Rank of a group to be the minimum cardinality of a set of generators, Let p be an Adian-Sirvanjan prime and let *p be the coproduct in B p' If G l , G 2 E Bp must Rank( G l *p G 2 ) Rank(G l ) + Rank(G2 )? Question 4: If p is an Adian-Sirvanjan prime and r 2: 2 is an integer must Fr(Bp) be Hopfian?
Question 5: If n is an Adian-Sirvanjan integer and H E Bn is finitley generated and universally equivalent to F 2 (B n ) must H be embeddable in some Fr(Bn)? Question 6: If n is an Adian-Sirvanjan integer and 2 ~ r < s ~ w must Fr(Bn) == Fs(Bn)?
8. References [A] S,l. Adian," Classification of periodic words and their application in group theory," Springer Lecture Notes in Mathematics 806, Burnside Groups, J,L, Mennicke, Editor, Springer-Verlag, Berlin (1980), 1 -40,
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[B] B. Baumslag,"Residually free groups," Proc. London Math. Soc. (3) 17 (1967), 402 - 418. [GB] G. Baumslag, "On generalised free products," Math. Z. 78 (1962), 423-438. [BMR] G.Baumslag, A.G. Myasnikov and V.N. Remeslennikov, "Algebraic geometry over groups I. Algebraic sets and ideal theory," J. Alg. 219 (1999), 16 - 79. [G] G. Gratzer, Universal Algebra, Van Nostrand, Princeton (1968). [GS] A.M. Gaglione and D. Spellman,"The persistence of universal formulae in free algebras," Bull. Austral. Math. Soc. 36 (1987), 11 - 17. [K] A.I. Kostrikin,"The Burnside problem," Izv. akad. Nauk. Ser. Mat. 23 (1959), 3 - 34. [KM1] O. Kharlampovich and A.G. Myasnikov,"Irreducible affine varieties over a free group: II Systems in quasi-quadratic triangular form and description of residually free groups," J. Alg. 200 (1998), 517 - 570. [KM2] O. Kharlampovich and A.G. Myasnikov,"Tarski's problem about the elementary theory of free groups has a positive solution," Electron. Res. Announc. Amer. Math. Soc. 4 (December 1998), 101 - 108. [KN] L.G. Kovacs and M.F. Newman,"Hanna Neumann's problems on varieties of groups," Springer Lecture Notes in Mathematics 372, Proc. Intemat. Conf. Theory of Groups, Canberra (1973), 417 - 431. [MKS] W.Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Dover, Mineola (2004). [N] H. Neumann, Varieties of Groups, Springer - Verlag, New York (1967). [Ne] P.M. Neumann," Splitting groups and projectives in varieties of groups," QJ Maths (Oxford) 18 (1967), 325 - 352. [R] V.N. Remeslennikov,"3-Free groups," Siberian J. Math. 30 (6) (1989), 153 - 157.
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[Se] Z. Sela," Diophantine geometry over groups VI: The elementary theory of a free group," GAFA, in press. lSi] V.L. Sirvanjan,"Imbedding of the group B(oo,n) in the group B(2, n)," Math. USSR-Jzv. 4 (1977), 181 - 199.
Changing generators in non free groups Richard Goldstein
Department of Mathematics, The University of Albany, Albany, NY 12222. USA E-mail: [email protected] Let Fn be a free group of rank n with basis X = {Xl, X2,'" ,x n }. If Y = {Yl, Y2, ... ,Yn} is any other set of generators for Fn then they form a basis for Fn since finitely generated free groups are Hopfian. In this paper which shall show that this fact characterizes finitely generated free groups.
Keywords: free group, generators, cut vertex
Introduction Let Fn be a free group of rank n and X ={Xl, X2,'" ,xn } be a basis for Fn. Let {Ul,U2,'" ,un} be a set of n freely reduced words in X. A standard fact is that {Ul' U2,'" ,un} is another basis for Fn if and only if the endomorphism where (Xi) = Ui for i = 1,2" " ,n is an isomorphism. One property of fg free groups is that they are Hopfian. This asserts that any endomorphism which is surjective is an isomorphism. Thus, we can conclude that {Ul,U2,'" ,un} is a basis for Fn if and only if they generate Fn. Finitely generated free groups can be characterized in many distinct ways, for example they freely act on trees, they are the fundamental group of a finite graph, etc. The purpose of this paper is to give another characterization, or more precisely to characterize what it means for a finitely generated group G to not be free. Main Theorem Let W be a freely reduced word in X, we say that w is primitive if we can find words {W2,'" ,wn } such that {W,W2,'" ,wn } is a basis for Fn. A classical construction associates to a freely reduced word W in X a finite graph, called the coinitial (star) graph of w, see for example. l If r is a finite graph and v is a vertex of r then v is said to be a cut vertex of r if its removal disconnects r. Whiteheads" Cut vertex theorem" ,2 asserts that if W is primitive, then the coinitial graph of w has a cut vertex. Moreover if 149
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the coinitial graph of w has a cut vertex and w is not a single generator or its inverse, then there exists an automorphism ¢ of Fn such that the length of ¢(w) is less than that of w. Thus by iteration if w is not primitive then there exists an automorphism 'IjJ of Fn such that 'IjJ(w) has no cut vertex. If v is a cyclically reduced word which starts with Xi to some positive power and the coinitial graph of v has no cut vertex then the word XiV also has no cut vertex since its coinitial graph contains the coinitial graph of v and thus XiV is non-primitive. It is essential for the coinitial graph of a non-primitive word v to have no cut vertex to guarantee that XiV is non-primitive. For example abab is not primitive in the free group F2 with basis {a, b} but a 2bab is primitive since {a 2bab, ab} generates F2 and is therefore a basis. Let G be a group of rank n which is not free. Let < Xl, X2, ... ,Xn Irl' r2, ... > be a presentation of G. Suppose that one of the relators say rl is a primitive element in Fn with basis {Xl, X2, ... ,xn }. Let ¢ be an automorphism of Fn such that ¢(rl) = Xl. Now < Xl, X2,'" ,xnlxl' ¢(r2), ¢(r3),'" > is another presentation of G. We can now eliminate Xl from the generating set and therefore G has rank less than n. Hence we may assume that in any presentation of G with n generators then no relator is primitive. Let ¢ be an automorphism of Fn such that ¢(rl) is cyclically reduced and has no cut vertex. Now G has a presentation of the form < Xl, X2,'" ,xnl¢(rl), ¢(r2),'" >. Without loss of generality let us assume that ¢(rl) starts with Xl to some positive power. Clearly {xI¢(rl),x2,'" ,xn } generates G and thus we have the following result. Theorem 0.1. Let G be a group of rank n which is not free and let X = {XI,X2,'" ,xn } be a set of generators of G. There exists a new set of generators for G, {UI,U2,'" ,un}, where Uj is a word in X such that {UI' U2,'" , un} is not a basis for F n , the free group with basis X. In particular they do not generate Fn. Moreover we can make UI be a non-primitive word in Fn. References 1. Lyndon, R., Schupp,
P. "Combinatorial Group Theory." Springer-Verlag, (1977). 2. Whitehead, J., H., C. "On certain sets of elements in a free group." Proc. London Math. Soc. 41, (1936), 48-56.
Matrix Completions over Principal Ideal Rings William H. Gustafson
Texas Tech University, Lubbock, Texas Donald W. Robinson
Brigham Young University, Provo, Utah R. Bruce Richter
University of Waterloo, Waterloo, Ontario William P. Wardlaw
U. S. Naval Academy, Annapolis, Maryland
Dedicated to Anthony M. Gaglione on his sixtieth birthday and to the memory of William H. Gustafson
Abstract We show that if A is a k x n matrix over a principal ideal ring R, with k < n, and if d is any element of the ideal generated by the k x k minors of A, then A forms the top k rows of an n x n matrix of determinant d. This parallels a 1981 result of Gustafson, Moore, and Reiner, and continues a program initiated by Hermite in 1849. Then we use these results to obtain an extension of a 1997 result of Richter and Wardlaw for good matrices.
1. Introduction
If A is a k x n matrix with k < n, the matrix completion problem intiated 151
152
by Hermite asks if A can be completed to an n x n matrix with prescribed determinant d. Gustafson, Moore, and Reiner, at the beginning of [5], give a brief summary of the history of the problem of completing a k x n matrix with k < n over certain commutative rings to an n x n matrix over the same ring with appropriate determinant. They also include references to some of the principal players in this program initiated by Hermite in 1849. In our contribution below, we show in Theorem 1 that principal ideal rings are among the rings over which this matrix completion is always possible. Theorem 2 states the relationships between six properties of a k x n matrix over a commutative ring. It extends a similar theorem in [9] by giving a best possible exposition of these relationships. Finally we show in Theorem 3 that if such completions are always possible over each ring in a given collection of rings, then they are also always possible over the unrestricted direct product of the collection. Throughout this paper, R will denote a commutative ring with identity. If A is a k x n matrix over R with k :S n, then Dk(A) denotes the ideal of R generated by the k x k sub determinants of A. We say that A has left block form if A is equivalent over R to a matrix E = [L 0], where L is a k x (k + 1) matrix over Rand 0 is the k x (n - k - 1) zero matrix over R. That is, there are matrices P E GL(k, R) and Q E GL(n, R) such that PAQ = [L 0]. Note that if k = n - 1 or k = n, the 0 block is missing and E = L; indeed, we can take A = E = L. 2. Results The following lemma was proved but not explicitly stated in [5], and was used to prove their main result. For the sake of completeness, we include a proof here.
Lemma. Let A be a k x n matrix over the commutative ring R with identity, let k < n, and let d E Dk(A). If A has left block form over R, then A enlarges to an n x n matrix A * over R whose determinant is d and whose top k rows form the matrix A. Proof. Let P E GL(k, R) and Q E GL(n, R) be such that PAQ = E = [L 0], where L is k x (k + 1). Clearly, Dk(A) = Dk(E) = Dk(L). Let j Cj = (_l)k+l+ det(L j ), where L j is the k x k submatrix of L obtained by deleting the jth column of L. Thus we can write d E Dk(A) as a linear combination d = L. ajcj = det(L *), where L * is the (k+ 1) x (k + 1) matrix
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obtained from L by adding [al a2 ... ak+l] as its last row. Let p = det(P) and q = det(Q), and multiply the last row of L* by the unit pq to obtain the matrix M* with det(M*) = pdq. Now let E* be the direct sum of M* and the (n - k - 1) x (n - k - 1) identity matrix In-k-l; thus,
E*
=
[~* In_Ok-J
= [;]
is an n x n matrix over R with det(E*) = pdq whose first k rows form the matrix E = P AQ. It follows that the matrix
A has det(A*)
° I n-°k- l ]E Q _l=[P-IEQ-1]=[A] FQ-l A'
*=[P-l
= d and
*
its first k rows form the matrix A.
D
Our first main result is Theorem 1. Suppose that R is either a Dedekind domain or a principal ideal ring, and that A is a k x n matrix over R with k < n. If d is any element of Dk(A), then there is an nxn matrix A* over R with determinant det(A*) = d whose first k rows form the matrix A.
Proof. In view of our lemma, we need only establish that A has a left block form over R for each of the two cases. When R is a Dedekind domain, Theorem 1 is the main result of [5], where they proved a lemma that every k x n matrix over a Dedekind domain R has a left block form over R. They comment that this lemma was established in a more general form by Levy [6] in 1972. When R is a principal ideal ring, W. C. Brown shows in [2, Thm. 15.24, p. 194], that every matrix over R has a Smith normal form. When A is k x n over R with k < n, its Smith normal form is a left block form for A ~R.
D
Since every principal ideal domain is also a Dedekind domain, Theorem 1 only extends the result of [5] when R is a principal ideal ring with nonzero divisors of zero. We were especially interested in the connection between Theorem 1 and the 1997 result [9] regarding good matrices. In [9], R was a commutative ring with identity and an r x n matrix A over R was defined to be left good if, for every vector x in RlXT, the ideal (xA) generated by the entries in the vector xA is the same as the ideal (x) generated by the entries of the
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vector x. Our lemma allows us to extend the Main Theorem of [9] to our second main result. Theorem 2. Consider the following statements about an r x n matrix A over the commutative ring R with identity.
(1) The rows of A extend to a basis of RIxn. (2) A can be enlarged to a matrix A* E GL(n, R). (3) A has a Smith normal form [Ir 0]. (4) A has a right inverse over R. (5) Dr(A) = R. (6) A is left good. Then (a) The statements (1), (2), and (3) are equivalent over any commutative ring R with identity. (b) The statements (4), (5), and (6) are equivalent over any commutative ring R with identity. (c) The statement (3) implies the statement (4) but in general they are not equivalent. (d) If A has left block form then all six statements are equivalent. Proof. Theorem 2 (a), (b), and (c) was proved in [9], except for the implications (2) =} (3) and (5) =} (4), and the fact that (4) ~ (3). The statement (2) means that there is an (n - r) x n matrix A' over R and an n x n matrix B* over R such that A*=
[~,]
and A * B* = I is the n x n identity matrix. But then it is clear that AB* = [Ir 0] is a Smith normal form for A. That is, (2) =} (3). The implication (5) =} (4) is immediate from [8, Cor. 1.28, p. 84]. However, for the sake of completeness we give the following elementary proof. If M is any m x n matrix over R and v = (CI, ... , cr ) a vector of column indices of M, so that 1 :::; Cj :::; n, we let M(v) denote the m x r submatrix of M whose jth column is the cjth column of M. It is easy to see that if In is the n x n identity matrix, then M(v) = M In(v). Now each r-subset {CI' ... ,cr } of {I, 2, .. , ,n} with 1 :::; CI < C2 < ... < Cr :::; n corresponds uniquely to a vector v = (CI, ... , Cr), and we can number these vectors (perhaps lexicographically) VI, V2, ... , VN with N = (~).
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Let dj = det(A j ) with Aj = A(vj). Then Dr(A) = R implies 1 = 'L. bjd j for scalars b1 , b2 , ... , bN in R. Now, for each j = 1, 2, ... , N, let B j be the n x r matrix B j = In(vj)Adj(A j ), and let B be the n x r matrix B = 'L.bjBj. Then AB
=
I)jABj
=
L
=
LbjAIn(Vj)Adj(Aj)
bjAjAdj(Aj )
=
L
bjdjIr
= Ir
and (4) A has a right inverse B. That is, (5) =} (4). The following example from [4], attributed to Kaplansky in [1, p. 7], shows that (4) =f? (3) in Theorem 2 (c), and hence the result of Theorem 2 (c) is the best possible. Let R be the ring of polynomials in x, y, z over the real numbers modulo the ideal generated by x 2 + y2 + z2 - 1. This is the ring of polynomial functions on the standard 2-sphere in 3-space. The 1 x 3 matrix A = [x y z] has a right inverse AT. If it had a Smith normal form [1 0 0], then there would be a matrix Q E GL(3, R) such that AQ = [1 0 0]. Assume such a Q exists with last column q = [f 9 hjT. Then A q = xf +yg+zh = 0 for all points on the 2-Sphere. Thus q provides a tangent vector field to the 2-sphere which, because of independence of the columns of Q, is never zero on the 2-sphere. But no such vector field exists, as is shown in [3, p. 70]. This contradiction shows (4) =f? (3). (In fact, the same argument shows directly that A does not have a left block form, since that would require an invertible Q with AQ = [u v 0].) This completes the proof of Theorem 2 (a), (b), and (c). To establish (d), we first observe that when r = n, the implication (4) =} (2) is a tautology. Then we use our lemma to show that (5) =} (2) when r < n and A has a left block form over R. It is clear from (5) that 1 E R = Dr(A). By our lemma, A can be enlarged to a matrix A* with determinant 1 when r < n. It is well known that a matrix over a commutative ring R with identity is invertible over R if and only if its determinant is a unit in R. (See [7, Thm. 50, p. 158].) Hence (5) =} (2). 0 We remark that if A is an (n - 1) x n matrix over R, then it is already in left block form, so statements (1) - (6) of Theorem 2 are equivalent. In particular, if A has a right inverse, then it extends to an n x n matrix which is invertible over R. (The latter was shown using an outer product argument in [9].) Recall that in the proof of Theorem 1 we observed that if R was either a principal ideal ring or a Dedekind domain, then every r x n matrix over
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R with r :::::: n had a left block form. Thus we have the following corollary to Theorem 2.
Corollary. If R is a principal ideal ring or a Dedekind domain, then statements (1) - (6) of Theorem 2 are equivalent. This corollary extends the Main Theorem of [9] from principal ideal rings to rings which are either principal ideal rings or are Dedekind domains. Our next theorem allows further extension of the class of rings for which certain properties mentioned above hold. Let R be a commutative ring with identity. Then R has property L if every r x n matrix A over R with r :::::: n has a left block form over R. R has property C if every r x n matrix A over R with r < n has, for each dE Dr(A) an n x n completion A* with det(A*) = d. R has property G if statements (1) - (6) of Theorem 2 are equivalent for every r x n matrix A over R with r :::::: n. Note that L =} C =} G, by our Lemma and Theorem 2. Theorem 3. Let P be anyone of the properties L, C, or G, and let R = EBjRj be the unrestricted direct sum of the commutative rings R j (j E J), where each R j has identity 1j. Then R has property P if and only if each R j has property P. Proof. We consider R to be an internal direct sum, so each R j is a subring and an ideal of R. For each a E R, aj = alj denotes the projection of a into Rj; we call aj the j-component of a. Thus (aj h = 0 if j =f. k and (aj)j = aj for all j, k E J. If A is a matrix over R, then we let Aj = ljA be the matrix of the same size over R j obtained by replacing each entry in A by its j-component. We write Ai to denote a matrix chosen with entries in R j , to distinguish it from the j-component Aj = ljA obtained from a matrix A already chosen with entries in R. In the proofs below, we will often define a matrix A over R by first specifying a matrix Aj over R j for each j E J, and letting A be the matrix of the same size over R with j-component Aj = ljA = Ai. Now suppose that R has property L and that Aj E (Rjyxn with r :::::: n. Since Aj E Rrxn, there are matrices P E GL(r, R) and Q E GL(n, R) such that PAjQ = E = [L 0], with L E w x (r+l). But Ai = ljA' implies that PAjQ = P(ljA')Q = (ljP)(Aj)(ljQ) = PjAjQj = E = E j = [L j 0] with Pj E GL(r, R j ) and Qj E GL(n, R j ). Thus, R j has property L.
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On the other hand, suppose that for each j E J, R j has property L, and that A E Rrxn with r n. Then Aj E (Rjyxn for each j E J, and so there are matrices Pj E GL(r, R j ) and Qj E GL(n, Rj) such that PjAjQj = [Lj Ol with Lj E (Rjyx(r+l). Let P E wxr be the matrix with j-component IjP = Pj and let Q E Rnxn be the matrix with j_ component IjQ = Qj for every j E J. It is easy to see that P E GL(r, R), Q E GL(n, R), and PAQ = [L Ol with L E Rrx(r+l) such that IjL = Lj for each j E J. Thus, R has property L. Now suppose that R has property C and that Aj E (Rj)rxn with r < n and d E Dr(A~). Since Aj E Rrxn, there is an A* E Rnxn whose first R rows form the matrix Aj and with determinant det(A*) = d. But the first R rows of IjA* = (A*)j E (Rj)nxn also form the matrix Aj and det((A*)j) = d = dj . Thus, R j has property C. On the other hand, suppose that for each j E J, R j has property C, A E Rrxn with r < n, and d E Dr(A). For each j E J, IjA = Aj E (Rjyxn has ljd = dj E Dr(Aj) and has an n x n completion (Aj)* over R j with det((Aj)*) = d = dj . Let A* be the n x n matrix over R with IjA* = (A*)j = (Aj)* for each j E J. Since det((A*)j) = dj for each j E J, it follows that det(A*) = d. Since the first r rows of (A*)j form the matrix Aj for each j E J, it follows that the first R rows of A * form the matrix A. That is, A* is the n x n completion of A with determinant d. Hence, R has property C. Suppose R has property G and that Aj, (Bj)T E (Rjyxn satisfy AjBj = (Ir)j, which is statement (4) of Theorem 3 for the ring R j . Let E = [Ir Ol- [Ir Olj, A = E+Aj, and B = ET +Bj. Note that A = [Ir 0li if i -=1= j, Aj = Aj, and similarly for B. Then AB = Ir shows that A satisfies (4) for the ring R. Since R has property G, A must also satisfy (2), so A has an invertible completion A* over R. It follows that (A*)j E GL(n,R j ) is the n x n completion of Aj = Aj over R j . Thus, (4) =} (2) in R j , so R j has property G. Finally, suppose for each j E J that R j has property G and that A, BT E R rxn satisfy AB = I r . Then AjBj = (Ir)j for each j E J, and so property G ensures that each Aj can be completed to an (Aj)* E GL(n, Rj ). Now let A* be the n x n matrix over R with j-component IjA* = (A*)j = (A j )*. Then A* E GL(n,R) and its first R rows form the matrix A. Thus (4) =} (2) in R, so R has property G. 0
s:
158
References 1. H. Bass, Introduction to some methods of algebraic K-theory, CBMS 20, Amer. Math. Soc., Providence, RI, 1974. 2. W. C. Brown, Matrices over Commutative Rings, Dekker, New York, 1992. 3. M. J. Greenberg, Lectures on Algebraic Topology, W. A. Benjamin, New York, 1967. 4. W. H. Gustafson, P. R. Halmos, and J. M. Zelmanowitz, The Serre Conjecture, Amer. Math. Monthly 85 (1978), 357-359. 5. W. H. Gustafson, M. E. Moore, and I. Reiner, Matrix completions over Dedekind rings, Linear and Multilinear Algebra 10 (1981), 141-144. 6. L. S. Levy, Almost diagonal matrices over Dedekind domains, Math. Z. 124 (1972), 89-99. 7. N. H. McCoy, Rings and Ideals, Mathematical Association of America, Washington, 1965. 8. B. R. McDonald, Linear Algebra over Commutative Rings, Dekker, New York, 1984. 9. R. B. Richter and W. P. Wardlaw, Good matrices: matrices which preserve ideals, Amer. Math. Monthly 104 (1997) , 932-938.
A primer on computational group homology and cohomology using GAP and SAGE David Joyner
Department of Mathematics, US Naval Academy, Annapolis, MD, [email protected].
Dedicated to my friend and colleague Tony Gaglione on the occasion of his sixtieth birthday These are expanded lecture notes of a series of expository talks surveying basic aspects of group cohomology and homology. They were written for someone who has had a first course in graduate algebra but no background in cohomology. You should know the definition of a (left) module over a (non-commutative) ring, what Z[G] is (where G is a group written multiplicatively and Z denotes the integers), and some ring theory and group theory. However, an attempt has been made to (a) keep the presentation as simple as possible, (b) either provide an explicit reference or proof of everything. Several computer algebra packages are used to illustrate the computations, though for various reasons we have focused on the free, open source packages, such as GAP [Gap] and SAGE [St] (which includes GAP). In particular, Graham Ellis generously allowed extensive use of his HAP [EI] documentation (which is sometimes copied almost verbatim) in the presentation below. Some interesting work not included in this (incomplete) survey is (for example) that of Marcus Bishop [Bi], Jon Carlson [C] (in MAGMA), David Green [Gr] (in C), Pierre Guillot [Gu] (in GAP, C++, and SAGE), and Marc Roder [Ro]. Though Graham Ellis' HAP package (and Marc Roder's add-on HAPcryst [RoJ) can compute comhomology and homology of some infinite groups, the computational examples given below are for finite groups only. 1. Introduction
First, some words of motivation. 159
160
Let G be a group and A a G-module a . Let A C denote the largest submodule of A on which G acts trivially. Let us begin by asking ourselves the following natural question. Question: Suppose A is a submodule of a G-module B and x is an arbitrary G-fixed element of BfA. Is there an element bin B, also fixed by G, which maps onto x under the quotient map? The answer to this question can be formulated in terms of group cohomology. ("Yes", if Hl(G, A) = 0.) The details, given below, will help motivate the introduction of group cohomology. Let Ac is the largest quotient module of A on which G acts trivially. Next, we ask ourselves the following analogous question. Question: Suppose A is a submodule of a G-module Band b is an arbitrary element of Bc which maps to 0 under the natural map Bc ---+ (B f A)c. Is there an element a in ac which maps onto b under the inclusion map? The answer to this question can be formulated in terms of group homology. ("Yes", if H1(G, A) = 0.) The details, given below, will help motivate the introduction of group homology. Group cohomology arises as the right higher derived functor for A t--------+ A c. The cohomology groups of G with coefficients in A are defined by
(See §4 below for more details.) These groups were first introduced in 1943 by S. Eilenberg and S. MacLane [EM]. The functor A t--------+ A C on the category of left G-modules is additive and left exact. This implies that if
is an exact sequence of G-modules then we have a long exact sequence of cohomology 0---+ AC---+Bc ---+ CC ---+ Hl(G,A)---+ Hl(G,B) ---+ Hl(G,C) ---+ H2(G,A) ---+ ...
(1)
aWe call an abelian group A (written additively) which is a left Z[G]-module a Gmodule.
161
Similarly, group homology arises as the left higher derived functor for A f------+ Ae. The homology groups of G with coefficients in A are defined by
Hn(G,A) = Tor~[el(Z,A). (See §5 below for more details.) The functor A f------+ Ae on the category of left G-modules is additive and right exact. This implies that if
is an exact sequence of G-modules then we have a long exact sequence of homology
H2(G,C) ----+ H1(G,A) ----+ H1(G,B)----+ Hl(G,C) ----+ Ae ----+ Be ----+ Ce ----+ o.
..• ----+
(2)
Here we will define both cohomology Hn(G, A) and homology Hn(G, A) using projective resolutions and the higher derived functors Ext n and Tor n. We "compute" these when G is a finite cyclic group. We also give various functorial properties, such as corestriction, inflation, restriction, and transfer. Since some of these cohomology groups can be computed with the help of computer algebra systems, we also include some discussion of how to use computers to compute them. We include several applications to group theory. One can also define Hl(G, A), H2(G, A), ... , by explicitly constructing co cycles and coboundaries. Similarly, one can also define HdG,A), H 2 (G,A), ... , by explicitly constructing cycles and boundaries. For the proof that these constructions yield the same groups, see Rotman [R], chapter 10. For the general outline, we follow §7 in chapter 10 of [R] on homology. For some details, we follow Brown [B], Serre [S] or Weiss [W]. For a recent expository account of this topic, see for example Adem [A]. Another good reference is Brown [B]. 2. Differential groups In this section cohomology and homology are viewed in the same framework. This "differential groups" idea was introduced by Cartan and Eilenberg [CE], chapter IV, and developed in R. Godement [G], chapter 1, §2. However, we shall follow Weiss [W], chapter 1.
162
2.1. Definitions A differential group is a pair (L, d), L an abelian group and d : L - t L a homomorphism such that d2 = O. We call d a differential operator. The group
H(L)
= Kernel (d)jlmage (d)
is the derived group of (L, d). If
then we call L graded. Suppose d (more precisely, diLJ satisfies, in addition, for some fixed r -I- 0,
We say d is compatible with the grading provided r = ±l. In this case, we call (L, d, r) a graded differential group. As we shall see, the case r = 1 corresponds to cohomology and the the case r = -1 corresponds to homology. Indeed, if r = 1 then we call (L, d, r) a (differential) group of cohomology type and if r = -1 then we call (L, d, r) a group of homology type. Note that if L = EB~=_ooLn is a group of cohomology type then L' = EB~_ooL~ is a group of homology type, where L~ = L-n' for all n E Z. For the impatient: For cohomology, we shall eventually take L = EBnHomc(Xn, A), where the Xn form a chain complex (with +1 grading) determined by a certain type of resolution. The group H(L) is an abbreviation for EBnExt Z[c] (Z, A). For homology, we shall eventually take L = EBnZ@Z[C] X n , where the Xn form a chain complex (with -1 grading) determined by a certain type of resolution. The group H(L) is an abbreviation for EBn Tor ~[C] (Z, A).
Let (L,d) = (L,dL) and (M,d) = (M,dM) be differential groups (to be more precise, we should use different symbols for the differential operators of Land M but, for notational simplicity, we use the same symbol and hope the context removes any ambiguity). A homomorphism f : L - t M satisfying do f = f 0 d will be called admissible. For any nEZ, we define nf : L - t M by (nf)(x) = n· f(x) = f(x) + .,. + f(x) (n times). If f
163
is admissible then so is nf, for any n E Z. An admissible map f gives rise to a map of derived groups: define the map f* : H(L) ~ H(M), by f*(x + dL) = f(x) + dM, for all x E L.
2.2. Properties Let
f be an admissible map as above.
(1) The map f* : H(L) ~ H(M) is a homomorphism. (2) If f : L ~ M and 9 : L ~ M are admissible, then so is f + 9 and we have (J + g)* = f* + g*. (3) If f : L ~ M and 9 : M ~ N are admissible then so is go f : L ~ N and we have (g 0 J)* = g* 0 f*. (4) If (3)
is an exact sequence of differential groups with admissible maps i, j then there is a homomorphism d* : H(N) ~ H(L) for which the following triangle is exact: H(L)
H(N)
/ (4)
H(M) This diagram b encodes both the long exact sequence of cohomology (1) and the long exact sequence of homology (2). Here is the construction of d*: Recall H (N) = Kernel (d) jlmage (d), so any x E H (N) is represented by an n E N with dn = O. Since j is surjective, there is an m E M bThis is a special case of TMoreme 2.1.1 in [G].
164
such that j(m) = n. Since j is admissible and the sequence is exact, j(dm) = d(j(m» = dn = 0, so dm E Kernel(j) = Image (i). Therefore, there is an £ E L such that dm = i(£). Define d*(x) to be the class of £ in H(L), i.e., d*(x) = £ + dL. Here's the verification that d* is well-defined: We must show that if we defined instead d* (x) = £' + dL, some £' E L, then £' - £ E dL. Pull back the above n E N with dn = 0 to an m E M such that j (m) = n. As above, there is an £ E L such that dm = i(£). Represent x E H(N) by an n' E N, so x = n' + dN and dn' = O. Pull back this n' to an m' E M such that j(m') = n'. As above, there is an £' E L such that dm' = i(£'). We know n' - n E dN, so n' - n = dn", some n" E N. Let j(m") = n", some m" E M, so j(m'-m-dm") = n' = n-j(dm") = n'-n-dj(m") = n'-n-dn" = O. Since the sequence L - M - N is exact, this implies there is an £0 E L such that i(£o) = m' - m - dm". But r~(£o) = i(d£o) = dm' - dm = ief') - i(£) = i(£' - f), so f' - £ E dL. (5) If M = L ffi N then H(M) = H(L) ffi H(N). proof: To avoid ambiguity, for the moment, let dx denote the differential operator on X, where X E {L,M,N}. In the notation of (3), j is projection and i is inclusion. Since both are admissible, we know that dMIL = d L and dMIN = dN. Note that H(X) C X, for any differential group X, so H(M) = H(M) n L ffi H(M) nNe H(L) ffi H(N). It follows from this that that d* = O. From the exactness of the triangle (4), it therefore follows that this inclusion is an equality.
o (6) Let L, L', M, M', N, N' be differential groups. If
o -----.
L ~M ~N - - 0
fl o -----.
.,
91
L' ~M'
hI
(5)
., -.L- N' - - 0
is a commutative diagram of exact sequences with i, i', j, j', j, g, h all admissible then
H(L) ~ H(M)
1·1
.,
9·1
H(L') ~ H(M')
165
commutes, j. H(M) ---. H(N)
9·1
h·l ./
'. H(N') H(M') ---. commutes, and d. H(N) ---. H(L)
1·1
h·l d.
H(N') ---. H(L') commutes. This is a case of Theorem 1.1.3 in [W] and of Theoreme 2.1.1 in [G]. The proofs that the first two squares commute are similar, so we only verify one and leave the other to the reader. By assumption, (5) commutes and all the maps are admissible. Representing x E H(M) by x = m + dM, we have
+ dN) = hj(m) + dN' = gi'(m) + dN' g*(i'(m) + dM') = g*i:(m + dM) = g*i:(x),
h*j*(x) = h*(j(m) =
as desired. The proof that the last square commutes is a little different than this, so we prove this too. Represent x E H(N) by x = n + dN with dn = 0 and recall that d*(x) = £+dL, where dm = i(£), £ E L, where j(m) = n, for m E M. We have
On the other hand,
d*h*(x)
= d*(h(n) + dN') = f!' + dL',
for some f!' E L'. Since h(n) EN', by the commutativity of (5) and the definition of d*, £' E L' is an element such that i'(£') = gi(£). Since i' is injective, this condition on £' determines it uniquely mod dL'. By the commutativity of (5), we may take f!' = J(£).
166
(7) Let L, L', M, M', N, N' be differential graded groups with grading +1 (i.e., of "cohomology type"). Suppose that we have a commutative diagram, with all maps admissible and all rows exact as in (5). Then the following diagram is commutative and has exact rows:
This is Proposition 1.1.4 in [W]. As pointed out there, it is an immediate consequence of the properties, 1-6 above. Compare this with Proposition 10.69 in [R]. (8) Let L, L', M, M', N, N' be differential graded groups with grading -1 (Le., of "homology type"). Suppose that we have a commutative diagram, with all maps admissible and all rows exact, as in (5). Then the following diagram is commutative and has exact rows: _d_.~
- j _ .~
Hn{N)
_j_~~
Hn(N') _d_.
H n _ 1 (L)
f. . __
~
H
+ (N') n 1
~
_d_.
HnCL')
_i~~
Hn(M')
~
J
Hn_1(L') _ _
~
This is the analog of the previous property and is proven similarly. Compare this with Proposition 10.58 in [R]. (9) Let (L, d) be a differential graded group with grading T. If dn = dl Ln then dn +r 0 dn = 0 and
(6) is exact. (10) If {Ln I nEil} is a sequence of abelian groups with homomorphisms d n satisfying (6) then (L, d) is a differential group, where L = EBnLn and d = EBndn.
2.3. Homology and cohomology When T = 1, we call Ln the group of n-cochains, Zn = Ln n Kernel (d n ) the group of n-cocycles, and Bn = Ln n dn-1(L n - 1) the group of ncoboundaries. We call Hn(L) = Zn/Bn the nth cohomology group. When T = -1, we call Ln the group of n-chains, Zn = LnnKernel (d n ) the group of n-cycles, and Bn = Ln ndn+ 1(L n+ 1) the group of n-boundaries. We call Hn(L) = Zn/Bn the nth homology group.
.
167
3. Complexes We introduce complexes in order to define explicit differential groups which will then be used to construct group (co)homology. 3.1. Definitions Let R be a non-commutative ring, for example R = Z[G]. We shall define a "finite free, acyclic, augmented chain complex" of left R-modules. A complex (or chain complex or R-complex with a negative grading) is a sequence of maps
... - ;
X n+l
0,,+1 -;
Xn
On
---t
Xn-
1
0,,-1 -;
X n-2
-; ...
(7)
for which OnOn+l = 0, for all n. If each Xn is a free R-module with a finite basis over R (so is ~ Rk, for some k) then the complex is called finite free. If this sequence is exact then it is called an acyclic complex. The complex is augmented if there is a surjective R-module homomorphism E : Xo -; Z and an injective R-module homomorphism f1. : Z -; X-I such that 00 = f1. 0 E, where (as usual) Z is regarded as a trivial R-module. The standard diagram for such an R-complex is . .. -------+
X2
82
-------+
X1
81
-------+
Xo
80
X -1
-------+
z ___
Z
1
ro
o
8_ 1
---+
X-
2 -------+ ...
Such an acyclic augmented complex can be broken up into the positive part
and the negative part
o -; ~ -;JJ. X -1 0-1 -; '71
X -2
0_2 -;
X -3
-; ...
Conversely, given a positive part and a negative part, they can be combined into a standard diagram by taking 00 = f1. 0 E.
168
If X is any left R-module, let X* = HomR(X, Z) be the dual Rmodule, where Z is regarded as a trivial R-module. Associated to any f E HomR(X, Y) is the pull-back f* E HomR(Y*, X*). (If y* E y* then define f* (y*) to be y* 0 f : X ---> Z.) Since "dualizing" reverses the direction of the maps, if you dualize the entire complex with a -1 grading, you will get a complex with a +1 grading. This is the dual complex. When R = Z[G] then we call a finite free, acyclic, augmented chain complex of left R-modules, a G-resolution. The maps Oi : Xi ---> X i - 1 are sometimes called boundary maps. Remark 3.1. Using the command BoundaryMap in the GAP CRIME package of Marcus Bishop, one can easily compute the boundary maps of a cohomology object associated to a G-module. However, G must be a p-group. Example 3.1. We use the package HAP [El] to illustrate some of these concepts more concretely. Let G be a finite group, whose elements we have ordered in some way: G = {9b ... , 9n}. Since a G-resolution X* determines a sequence of finitely generated free Z[G]-modules, to concretely describe X* we must be able to concretely describe a finite free Z[G]-module. In order to represent a word w in a free Z[G]-module M of rank n, we use a list of integer pairs w = [[i 1,el],[i 2,e2], ... ,[ik,ek]]. The integers ij lie in the range {-n, ... ,n} and correspond to the free Z[G]-generators of M and their additive inverses. The integers ej are positive (but not necessarily distinct) and correspond to the group element gej' Let's begin with a HAP computation.
r---------------------------GAP--------------------______~ gap> LoadPackage ("hap") ;
true gap> gap>
G:~Group
([ (1,2,3), (1,2) 1);;
R:~Reso1utionFiniteGroup(G,
4);;
This computes the first 5 terms of a G-resolution (G
= 83)
The bounday maps 8i are determined from the boundary component of the GAP record R. This record has (among others) the following components:
• R! .dimension(k) - the Z[G]-rank of the module X k ,
169 • R! . boundary(k, j) - the image in Xk-l of the j-th free generator of
Xk, • R! . elts - the elements in G, • R! . group is the group in question.
Here is an illustration:
r----------------------------
GAP ----------------------------~
gap> R! .group; Group([ (1,2), (1,2,3) ]) gap> R! .elts; [ (), (2,3), (1,2), (1,2,3), (1,3,2), (1,3) ] gap> R! .dimension(3); 4 gap> R! .boundary(3,1); [ [ 1, 2 ], [ -1, 1 ] gap> R! .boundary(3,2); [ [ 2, 2 ], [ -2, 4 ] gap> R! .boundary(3,3); [ [ 3, 4 ], [ 1, 3 ], -3, 1 ], -1, 1 ] ] gap> R! .boundary(3,4); [ [ 2, 5 ], [ -3, 3 ], [ 2, 4 ], -1, 4 ], [ 2,
1 ],
[ -3, 1 ] ]
In other words, X3 is rank 4 as a G-module, with generators {iI, 12, 13, f4} say, and
Now, let us create another resolution and compute the equivariant chain map between them. Below is the complete GAP session:
r-----------------------------
GAP ------------------------------
gap> G1 :=Group ([ (1,2,3), (1,2) ]); Group([ (1,2,3), (1,2) ]) gap> G2 :=Group ([ (1,2,3), (2,3) ]); Group([ (1,2,3), (2,3) ]) gap> phi: =GroupHomomorphismBylmages (G1, G2, [ (1,2,3) , (1,2) ], [ (1,2,3) , (2,3) ] ) ; [ (1,2,3), (1,2) ] -> [ (1,2,3), (2,3) ] gap> R1:=ResolutionFiniteGroup(G1, 4); Resolution of length 4 in characteristic 0 for Group ([ (1,2), (1,2,3) ]) gap> R2:=ResolutionFiniteGroup(G2, 4); Resolution of length 4 in characteristic 0 for Group ([
(2,3),
(1,2,3)
])
.
170
gap> ZP_map:=EquivariantChainMap(Rl, R2, phi); Equivariant Chain Map between resolutions of length 4 . gap> map := TensorWithlntegers( ZP_map); Chain Map between complexes of length 4 . gap> [ fL gap> [ 2, gap> gap>
Hphi := Homology( map, 3); f2, f3 1 -> [ f2, f2*f3, fl*f2-2 Abelianlnvariants(Image(Hphi»; 3 1 GroupHomology(Gl,3);
[ 6 1 gap> GroupHomology(G2,3);
[ 6 1
In other words, H (1)) is an isomorphism (as it should be, since the homology is independent of the resolution choosen).
3.2. Constructions Let R
= Z[G].
3.2.1. Bar resolution This section follows §1.3 in [W]. Define a symbol [.] and call it the empty cell. Let Xo = R[.], so Xo is a finite free (left) R-module whose basis has only 1 element. For n > 0, let g1, ... , gn E G and define an n-cell to be the symbol [g1, ... , gn]. Let
where the sum runs over all ordered n-tuples in Gn. Define the differential operators d n and the augmentation module maps, by
E,
as G-
171
f(g[.]) = 1,
9 E
G
d 1 ([g]) = g[.]- [.], d 2([gl,g2])
= gl[g2]- [glg2] + [gl], n-l
dn ([gl, ... , gn])
= gl [g2, ... , gn] + I: (-I)i[gl, ... , gi-l, gigi+1, gi+2, ... , gn] i=1
+ (-I)n[gl,'"
,gn-l],
for n ~ 1. Note that the condition f(g[.]) = 1 for all 9 EGis equivalent to saying f([.J) = 1. This is because f is a G-module homomorphism and Z is a trivial G-module, so f(g[.]) = gf([.]) = 9 . 1 = 1, where the (trivial) G-action on Z is denoted by a '. The Xn are finite free G-modules, with the set of all n-cells serving as a basis. Proposition 3.1. With these definitions, the sequence
... -+
X2
d2 -+
X1
d -+ 1
X0
€ -+
'71 til
-+
0,
is a free G-resolution.
Sometimes this resolution is called the bar resolutionc . There are two other resolutions we shall consider. One is the closely related "homogeneous resolution" and the other is the "normalized bar resolution". This simple-looking proposition is not so simple to prove. First, we shall show it is a complex, Le., d 2 = O. Then, and this is the most non-trivial part of the proof, we show that the sequence is exact. First, we need some definitions and a lemma. Let f : L -+ M and 9 : L -+ M be +1-graded admissible maps. We say f is homotopic to 9 if there is a homomorphism D : L -+ M, called a homotopy, such that • Dn = DILn : Ln -+ M n+ 1 , • f - 9 = Dd + dD. CThis resolution is not the same as the resolution computed by HAP in Example 3.1. For details on the resolution used by HAP, please see Ellis [E2J.
172
If L = M and the identity map 1 : L -> L is homotopic to the zero map o : L -> L then the homotopy is called a contracting homotopy for L. Lemma 3.1. If L has a contracting homotopy then H(L)
= o.
proof: Represent x E H(L) by I! E L with dl! = O. But I! = 1 (I!) -O(I!) = + Dd(£) = dD(I!). Since D : L -> L, this shows I! E dL, so x = 0 in H(L).D Next, we construct a contracting homotopy for the complex X* in Proposition 3.1 with differential operator d. Actually, we shall temporarily let X-I = Z, X-n = 0 and d_ n = 0 for n > 1, so that that the complex is infinite in both directions. We must define D : X -> X such that
dD(£)
• • • • •
D-I = Dlz : Z -> X o, Dn = Dlxn : Xn -> X n+l , eD_ I = 1 on Z, dIDo + D_Ie = 1 on X o, dn+IDn + Dn-Idn = 1 in X n , for n 2: 1.
Define
n> 1, D_ I (1) = [.],
Do(g[.]) = [g], Dn(g[gl, ... ,gn]) = [g,gl, ... ,gn]' and extend to a Z-basis linearly. Now we must verify the desired properties. By definition, for m E Z, eD_I(m) = e(m[.]) eD_ I is the identity map on Z. Similarly,
n>O,
= me([.]) = m.
(dIDo + D-Ie)(g[.]) = dl([g]) + D_ I (1) = g[.]- [.] + D_ 1 (1) = g[.]- [.] + [.] = g[.]. For the last property, we compute
Therefore,
173
dn+lDn (g[gl, ... ,gn])
=
d n+l([g,gl,'" ,gn])
=
g[gl, ... ,gn]- [ggl,'" ,gn] n-l + _l)i-l [g, gl, ... ,gi-l, gigi+l, gi+2, ... ,gn]
2:) i=1
and
D n - 1 d n (g[gl, ... ,gn])
= D n - 1(gd n ([gl, ... ,gn])) = D n - 1 (ggl[g2, ... ,gn] n-l
+ 2) -l)i g [gl, ... , gi-l, gigi+l, gi+2,···, gn] i=1
+ (_l)ng[gl"'"
gn-l])
= [ggl,g2,'" ,gn] n-l + 2:)-l)i[g,gl,'" ,gi-l,gigi+l,gi+2,··. ,gn] i=1
+ (_1)n[g,gl,'"
,gn-l].
All the terms but one cancels, verifying that dn+1D n + D n - 1dn = 1 in X n , ~ 1. Now we show d 2 = O. One verifies d 1 d 2 = 0 directly (which is left to the reader). Multiply d k D k - 1 + Dk-2dk-l = 1 on the right by d k and d k+1 D k + Dk- 1 d k = 1 on the left by d k :
for n
dkD k- 1 d k + Dk-2dk-ldk
= d k = dkdk+1 D k + dkD k- 1d k· Cancelling like terms, the induction hypothesis dk-ldk = 0 implies d k dk+l = O. This shows d 2 = 0 and hence that the sequence in Proposition 3.1 is exact. This completes the proof of Proposition 3.1. 0 The above complex can be "dualized" in the sense of §3.1. This dualized complex is of the form
o --+
'7l tfJ
M
--+
X -1
d- 1 --+
X -2
d-2 --+
X -3
--+ ..•
The standard G-resolution is obtained by splicing these together.
174
3.2.2. Normalized bar resolution
Define the normalized cells by
*_{[91, ... ,gnJ, if allgi =/:-1,
[gl,···,gn J -
O 'f , 1 some gi
= 1.
Let Xo = R[.J and n 2': 1,
where the sum runs over all ordered n-tuples in Gn. Define the differential operators dn and the augmentation map exactly as for the bar resolution. Proposition 3.2. With these definitions, the sequence
••• ---->
X2
d2
---->
X1
d1 ---->
Xo
€
'77
----> u... ---->
0,
is a free G -resolution.
Sometimes this resolution is called the normalized bar resolution. proof: See Theorem 10.117 in [RJ. 0 3.2.3. Homogeneous resolution
Let Xo = R, so Xo is a finite free (left) R-module whose basis has only 1 element. For n > 0, let Xn denote the Z-module generated by all (n + 1)tuples (gO,.'" gn)· Make Xi into a G-module by defining the action by g: Xn ----> Xn by 9 : (gO, ... , gn)
I--->
(ggo, . .. , ggn),
9 E
G.
Define the differential operators an and the augmentation c, as Gmodule maps, by c(g) = 1, n-l
an (go, .,. ,gn) = 2:(-I)i(gO,'" , 9i-l,[Ji,gHl, ... ,gn), i=O
for n 2': 1. Proposition 3.3. With these definitions, the sequence
175
••• ----t
X2
02
----t
X1
01
----t
Xo
€ ----t
Z
----t
0,
is a G-resolution.
Sometimes this resolution is called the homogeneous resolution. Of the three resolutions presented here, this one is the most straightforward to deal with. proof: See Lemma 10.114, Proposition 10.115, and Proposition 10.116 in [R]. 0 4. Definition of Hn(G, A) For convenience, we briefly recall the definition of Ext n. Let A be a left R-module, where R = Z[G], and let (Xi) be a G-resolution of Z. We define Ext z[C](Z, A)
= Kernel (d~+l)/Image (d~),
where d~:
Hom(Xn_1,A)
----t
Hom(Xn,A),
is defined by sending f : X n - 1 ----t A to fd n : Xn ----t A. It is known that this is, up to isomorphism, independent of the resolution choosen. Recall Ext i[c] (Z, A) is the right-derived functors of the right-exact functor A f-----+ AC = Homc(Z, A) from the category of G-modules to the category of abelian groups. We define
(8) When we wish to emphasize the dependence on the resolution choosen, we write Hn(G,A,X*). For example, let X* denote the bar resolution in §3.2.1 above. Call C n = Cn(G, A) = Homc(Xn , A) the group of n-cochains of G in A, zn = zn(G, A) = C n n Kernel (8) the group of n-cocycles, and Bn = Bn(G,A) = 8(C n - 1) the group of n-coboundaries. We call Hn(G,A) = zn / B n the nth cohomology group of G in A. This is an abelian group. We call also define the cohomology group using some other resolution, the normalized bar resolution or the homogeneous resolution for example. If we wish to express the dependence on the resolution X* used, we write
176
Hn(G, A, X*). Later we shall see that, up to isomorphism, this abelian group is independent of the resolution. The group H 2 ( G, Z) (which is isomorphic to the algebraic dual group of H2(G,C X )) is sometimes called the Schur multiplier of G. Here C denotes the field of complex numbers. We say that the group G has cohomological dimension n, written cd(G) = n, if Hn+l(H,A) = 0 for all G-modules A and all subgroups H of G, but Hn(H, A) =I- 0 for some such A and H. Remark 4.1. • If cd( G) < 00 then G is torsion-freed. • If G is a free abelian group of finite rank then cd(G) = rank(G). • If cd( G) = 1 then G is free. This is a result of Stallings and Swan (see for example [RJ, page 885). 4.1. Computations
We briefly discuss computer programs which compute cohomology and some examples of known computations.
4.1.1. Computer computations of cohomology GAP [Gap] can compute some cohomology groupse. All the SAGE commands which compute group homology or cohomology require that the package HAP be loaded. You can do this on the command line from the main SAGE directory by typingf
sage -i gap_packages-4.4.10_3.spkg Example 4.1. This example uses SAGE, which wraps several of the HAP functions . .-----______________________ SAGE
I
sage: G = AlternatingGroup(5)
dThis follows from the fact that if G is a cyclic group then Hn(G,7l..) i= 0, discussed below. eSee §37.22 of the GAP manual, M. Bishop's package CRIME for cohomology of p-groups, G. Ellis' package HAP for group homology and cohomology of finite or (certain) infinite groups, and M. Roder's HAPCryst package (an add-on to the HAP package). SAGE [Stl computes cohomology via it's GAP interface. fThis is the current package name - change 4.4.10_3 to whatever the latest version is on http://www.sagemath.org/packages/optional/atthetimeyoureadthis.Also.this command assumes you are using SAGE on a machine with an internet connection.
177 sage: G.cohomology(l,7) Trivial Abelian Group sage: G.cohomology(2,7) Trivial Abelian Group
4.1.2. Examples Some example computations of a more theoretical nature.
(1) HO(G,A)
=
AG.
This is by definition. (2) Let L/ K denote a Galois extension with finite Galois group G. We have Hl(G,LX) = 1. This is often called Hilbert's Theorem 90. See Theorem 1.5.4 in [W] or Proposition 2 in §X.1 of [S]. (3) Let G be a finite cyclic group and A a trivial torsion-free G-module. Then Hl(G,A) = O. This is a consequence of properties given in the next section. (4) If G is a finite cyclic group of order m and A is a trivial G-module then
This is a consequence of properties given below. For example, H2(GF(q)X,C) = O. (5) If IGI = m, rA = 0 and gcd(r, m) = 1, then Hn(G, A) = 0, for all n~1.
This is Corollary 3.1.7 in [W]. For example, H 1 (A 5 ,7L/77L) = O.
5. Definition of Hn(G, A) We say A is projective if the functor B f----' HomG(A, B) (from the category of G-modules to the category of abelian groups) is exact. Recall, if P.Z = ...
-->
P2
d2
-->
is a projective resolution of 7L then
P1
d1 n € '71 --> r-o --> ~ -->
0
(9)
178
It is known that this is, up to isomorphism, independent of the resolution choosen. Recall Tor ~[c] (Z, A) are the right-derived functors of the rightexact functor A ~ Ac = Z 0z[c] A from the category of G-modules to
the category of abelian groups. We define
(10) When we wish to emphasize the dependence on the resolution, we write Hn(G, A, Pz).
Remark 5.1. If G is a p-group, then using the command ProjectiveResolution in GAP's CRIME package, one can easily compute the minimal projective resolution of a G-module, which can be either trivial or given as a MeatAxe g module. Since we can identify the functor A ~ Ac with A ~ A0z[c] Z (where Z is considered as a trivial Z[G]-module), the following is another way to formulate this definition. If Z is considered as a trivial Z[G]-module, then a free Z[G]-resolution of Z is a sequence of Z[G]-module homomorphisms
satisfying: • (Freeness) Each Mn is a free Z[G]-module. • (Exactness) The image of Mn+1--Mn equals the kernel of Mn--Mn-l for all n > O. • (Augmentation) The cokernel of Ml--Mo is isomorphic to the trivial Z[G]-module Z. The maps Mn --Mn-l are the boundary homomorphisms of the resolution. Setting TMn equal to the abelian group Mn/G obtained from Mn by killing the G-action, we get an induced sequence of abelian group homomorphisms
... --TMn--TMn_l-- ... --TM1--TMo This sequence will generally not satisfy the above exactness condition, and one defines the integral homology of G to be gSee for example http://wvw.math.rwth-aachen.de/-MTX/.
179
Hn(G,Z) = Kernel(TMn~TMn_l)/Image(TMn+l~TMn) for all n
> o.
5.1. Computations We briefly discuss computer programs which compute homology and some examples of known computations.
5.1.1. Computer computations of homology Example 5.1. GAP will compute the Schur multiplier H 2 (G, Z) using the AbelianlnvariantsMultiplier command. To find H 2 (A 5,Z), where A5 is the alternating group on 5 letters, type . -_____________________________ GAP ____________________________--, gap> AS:=AlternatingGroup(S); Alt ( [ 1 .. 5 J ) gap> AbelianlnvariantsMultiplier(AS); [ 2 J
So, H 2 (A 5 , q ~ Z/2Z. Here is the same computation in SAGE: . -__________________________ SAGE sage: G = AlternatingGroup(S) sage: G.homology(2) Multiplicative Abelian Group isomorphic to C2
Example 5.2. The SAGE command poincare_series returns the Poincare series of G (mod p) (p must be a prime). In other words, if you input a (finite) permutation group G, a prime p, and a positive integer n, poincare_series(G,p,n) returns a quotient of polynomials f(x) = P(x)/Q(x) whose coefficient of xk equals the rank of the vector space Hk(G, ZZ/pZZ) , for all k in the range 1 ::; k ::; n . r -____________________________
sage: sage: (x'2 sage: sage:
G = SymmetricGroup(S) G.poincare_series(2,10) + 1)/(x'4 - x'3 - x + 1) G = SymmetricGroup(3) G.poincare_series(2,lO)
SAGE
180 1/ (-x + 1)
This last one implies
for 1 ::; k ::; 10.
Example 5.3. Here are some more examples using SAGE's interface to HAP: . -__________________________ SAGE sage: G = SymmetricGroup(S) sage: G.homology(l) Multiplicative Abelian Group sage: G.homology(2) Multiplicative Abelian Group sage: G.homology(3) Multiplicative Abelian Group sage: G.homology(4) Multiplicative Abelian Group sage: G.homology(S) Multiplicative Abelian Group sage: G.homology(6) Multiplicative Abelian Group sage: G.homology(7) Multiplicative Abelian Group
isomorphic to C2 isomorphic to C2
isomorphic to C2 x C4 x C3 isomorphic to C2
isomorphic to C2 x C2 x C2 isomorphic to C2 x C2 isomorphic to C2 x C2 x C4 x C3 x CS
The last one means that
(Z/2Z)2 x (Z/3Z) x (Z/4Z) x (Z/5Z). r - - - - -________________________
SAGE
sage: G = AlternatingGroup(S) sage: G.homology(l) Trivial Abelian Group
sage: G.homology(1,7) Trivial Abelian Group sage: G.homology(2,7) Trivial Abelian Group
5.1.2. Examples Some example computations of a more theoretical nature.
181
(1) If A is a G-module then Tor~[G](Z,A) = Ho(G,A) = AG ~ AIDA. proof: We need some lemmas. Let to : Z[G] ----t Z be the augmentation map. This is a ring homomorphism (but not a G-module homomorphism). Let D = Kernel (to) denote its kernel, the augmentation ideal. This is a G-module. Lemma 5.1. As an abelian group, D is free abelian generated by G1 = {g - 1 I 9 E G}. We write this as D = Z(G - 1). proof of lemma: If d E D then d = L9EG mgg, where mg E Z and LgEG mg = O. Thus, d = L9EG mg(g - 1), so D c Z(G - 1). To show D is free: If L9EG mg(g - 1) = 0 then L9EG mgg - L9EG mg = 0 in Z[G]. But Z[G] is a free abelian group with basis G, so mg = 0 for all 9 E G. 0 Lemma 5.2. Z 0z[G] A = AIDA, where DA is generated by elements of the form ga - a, 9 E G and a E A. Recall AG denotes the largest quotient of A on which G acts triviallyh. proof of lemma: Consider the G-module map, A ----t Z0z[G]A, given by a f-------> 10a. Since Z0Z[G] A is a trivial G-module, it must factor through A G. The previous lemma implies AG ~ AIDA. (In fact, the quotient map q : A ----t AG satisfies q(ga - a) = 0 for all 9 E G and a E A, so DA C Kernel (q). By maximality of A G , DA = Kernel (q). QED) SO, we have maps A ----t AG ----t Z 0z[G] A. By the definition of tensor products, the map Z x A ----t A G, 1 x a f-------> 1 . aDA, corresponds to a map Z0z[G] A ----t AG for which the composition AG ----t Z0z[G] A ----t AG is the identity. This forces AG ~ Z 0z[G] A. 0 See also # 11 in §6. (2) If G is a finite group then Ho(G, Z) = Z. This is a special case of the example above (taking A = Z, as a trivial G-module). (3) H1(G,Z) ~ G/[G,G], where [G,G] is the commutator subgroup of G. This is Proposition 10.110 in [R], §10.7. proof: First, we claim: DI D2 ~ G/[G, G], where D is as in Lemma 5.l. To prove this, define G ----t DI D2 by 9 f-------> (g-1)+D2. Since gh-1(g-l) - (h-1) = (g-1)(h-1), it follows that e(gh) = e(g)e(h), so e is
e:
hImplicit in the words "largest quotient" is a universal property which we leave to the reader for formulate precisely.
182
a homomorphism. Since D / D2 is abelian and G / [G, Gj is the maximal abelian quotient of G, we must have Kernel (£I) c [G, Gj. Therefore, £I factors through £I': G/[G,Gj-t D/D2, g[G,G] 1-+ (g-l) +D2. Now, we construct an inverse. Define T : D -t G /[G, G] by 9 - 1 1 - + g[G, G]. Since T(g-l+h-l) = g[G, G]·h[G, G] = gh[G,Gj, it is not hard to see that this is a homomorphism. We would be essentially done (with the construction of the inverse of £I', hence the proof of the claim) if we knew D2 C Kernel (T). (The inverse would be the composition of the quotient D/D2 -t D/Kernel(T) with the map induced from T, D/Kernel(T)-t G/[G,G].) This follows from the fact that any x E D2 can be written as x = (2: g mg(g - 1»(2:h m',,(h - 1» = (2: g,h mgm',,(g - 1)(h - 1», so T(X) = I1 g, h(ghg-lh-l)mgm~[G,Gj = [G,G]. QED (claim) Next, we show H 1 (G,Z) ~ D/D2. From the short exact sequence
o -t D -t Z[Gj -..:. Z -t 0, we obtain the long exact sequence of homology ... -t H 1 (G,D) -t Hl(G,Z[G])-t
H 1 (G,Z)!-; Ho(G,D)!.. Ho(G,Z[G]) ~ Ho(G,Z) -t O.
(11)
a
Since Z[Gj is a free Z[Gj-module, H 1 (G, Z[G]) = O. Therefore is injective. By item # 1 above (i.e., Ho(G,A) ~ A/DA ~ Ae, we have Ho(G,Z) ~ Ze = Z and Ho(G,Z[G]) ~ Z[G]jD ~ Z. By (11), E* is surjective. Combining the last two statements, we find Z/Kernel (E*) ~ Z.This forces E* to be injective. This, and (11), together imply f must be O. Since this forces to be an isomorphism, we are done. 0 (4) Let G = F/R be a presentation of G, where F is a free group and R is a normal subgroup of relations. Hopf's formula states: H 2 ( G, Z) ~ (F n R)/[F, RJ, where [F, R] is the commutator subgroup of G. See [RJ, §1O.7. The group H 2(G,Z) is sometimes called the Schur multiplier of G.
a
6. Basic properties of Hn(G, A), Hn(G, A)
Let R be a (possibly non-commutative) ring and A be an R-module. We say A is injective if the functor B 1-+ Home(B, A) (from the category of Gmodules to the category of abelian groups) is exact. (Recall A is projective if the functor B 1-+ Home(A, B) is exact.) We say A is co-induced if it has the form Homz(R, B) for some abelian group B. We say A is relatively
183
injective if it is a direct factor of a co-induced R-module. We say A is relatively projective if 7r :
Z[G]®z A""""", A, x 121 a f------> xa,
maps a direct factor of Z[G]®z A isomorphically onto A. These are the Gmodules A which are isomorphic to a direct factor of the induced module Z[G]®zA. When G is finite, the notions of relatively injective and relatively projective coincide i . (1) The definition of Hn(G,A) does not depend on the G-resolution X* of Z used. (2) If A is an projective Z[G]-module then Hn(G, A) = 0, for all n 2: l. This follows immediately from the definitions. (3) If A is an injective Z[G]-module then Hn(G, A) = 0, for all n 2: l. See also [S], §VII.2. (4) If A is a relatively injective Z[G]-module then Hn(G,A) = 0, for all
n2:l. This is Proposition 1 in [S], §VII.2. (5) If A is a relatively projective Z[G]-module then Hn(G,A) = 0, for all
n2:l. This is Proposition 2 in [S], §VII.4. (6) If A = A' EB A" then Hn(G,A) = Hn(G,A') EB Hn(G,A"), for all n 2: O. More generally, if I is any indexing family and A = EBiEI Ai then Hn(G,A) = EBiEIHn(G, Ai), for all n 2: O. This follows from Proposition 10.81 in §10.6 of Rotman [R]. (7) If
is an exact sequence of G-modules then we have a long exact sequence of cohomology (1). See [S], §VII.2, and properties of the ext functor [R], §10.6. (8) A f------> Hn(G, A) is the higher right derived functor associated to A f------> AC = Homc(A, Z) from the category of G-modules to the category of abelian groups. This is by definition. See [S], §VII.2, or [R], §1O.7. iThese notions were introduced by Hochschild [Ho].
184
(9) If
is an exact sequence of G-modules then we have a long exact sequence of homology (2). In the case of a finite group, see [S], §VIII.1. In general, see [S], §VII.4, and properties of the Tor functor in [R], §1O.6. (10) A ~ Hn(G, A) is the higher left derived functor associated to A ~ AG = Z 0z[G] A on the category of G-modules. This is by definition. See [S], §VII.4, or [RJ, §10.7. (11) If G is a finite cyclic group then
Ha(G, A) = A G , H 2n - 1 (G, A) = A G jN A, H2n(G,A) = Kernel (N)jDA, for all n ~ 1. To prove this, we need a lemma. Lemma 6.1. Let G = (g) be acyclic group of order k. Let M and N = 1 + 9 + g2 + ... + gk-l. Then
..• --+
Z[G]
N
--+
Z[G]
M
--+
Z[G]
--+
Z[G]
N
--+
Z[G]
M
--+
Z[G]
€
--+
Z
=9-
--+
1
0,
is a free G-resolution. proof of lemma: It is clearly free. Since MN = NM = (g - 1)(1 + 9 + g2 + ... + gk-l) = gk _ 1 = 0, it is a complex. It remains to prove exactness. Since Kernel (€) = D = Image (M), by Lemma 5.1, this stage is exact. To show Kernel (M) = Image (N), let x = L7':~ mj gj E Kernel (M). Since (g - l)x = 0, we must have ma = ml = ... = mk-l. This forces x = maN E Image (N). Thus Kernel (M) C Image (N). Clearly M N = 0 implies Image (N) C Kernel (M), so Kernel (M) = Image (N). To show Kernel (N) = Image (M), let x = L~;:~ mjgJ E Kernel (N). Since Nx = 0, we have 0 o. Observe that
= €(Nx) = €(N)€(x) = k€(x), so
L;;:~ mj
=
185
x
= ma' 1 + mIg + m2g 2 + ... + mk_lg k- l = (ma - mag) + (ma + ml)g + m2g 2 + ... + mk_lg k- l = (ma - mag) + (ma + ml)g - (ma + ml)g2 +(ma + ml + m2)g2 - (ma + ml + m2)g3 + ... +(ma + .. + mk_l)gk-l - (ma + .. + mk_l)gk.
where the last two terms are actually O. This implies x = -M(ma + (ma+mt)g+(ma+ml +m2)g2+ ... +(ma+ .. +mk_t)gk-1 E Image (M). Thus Kernel (N) C Image (M). Clearly N M = 0 implies Image (M) C Kernel (N), so Kernel (N) = Image (M). This proves exactness at every stage.D Now we can prove the claimed property. By property 1 in §5.1.2, it suffices to assume n > O. Tensor the complex in Lemma 6.1 on the right with A: ... --->
Z[G] @Z[G) A ~ Z[G] @Z[G) A ~ Z[G] @Z[G) A ~ Z[G] @Z[G) A ~ Z[G] @Z[G) A-=' Z @Z[G]A
--->
0,
where the new maps are distinguished from the old maps by adding an asterisk. By definition, Z[G] ®Z[G] A ~ A, and by property 1 in §5.1.2, Z ®Z[G] A ~ AIDA. The above sequence becomes
This implies, by definition of Tor,
and
Tor~~Gl(Z, A) = Kernel (N*)/Image (M*) = A[N]I DA. See also [S], §VIII.4.1 and the Corollary in §VIII.4. (12) The group H2(G, A) classifies group extensions of A by G. This is Theorem 5.1.2 in [W]. See also §10.2 in [R]. (13) If G is a finite group of order m = IGI then mHn(G, A) = 0, for all This is Proposition 10.119 in [R]. (14) If G is a finite group and A is a finitely-generated G-module then Hn( G, A) is finite, for all n 2: 1. This is Proposition 3.1.9 in [W] and Corollary 10.120 in [R].
186
(15) The group Hl(G, A) constructed using resolutions is the same as the group constructed using 1-cocycles. The group H2(G, A) constructed using resolutions is the same as the group constructed using 2-cocycles. This is Corollary 10.118 in [R]. (16) If G is a finite cyclic group then
HO(G,A) 2n H - 1 (G,A) H2n(G,A)
=
AC,
=
Kernel NIDA,
=
A CINA,
for all n :::: 1. Here N : A --+ A is the norm map N a = L9EC ga and DA is the augmentation ideal defined above (generated by elements of the form ga - a). proof: The case n = 0: By definition, HO(G,A) = Ext~[Cl(Z,A) = Homc(Z,A). Define T: Homc(Z,A) --+ AC by sending f f-----+ f(l). It is easy to see that this is well-defined and, in fact, injective. For each a E AC, define f = fa E Homc(Z,A) by f(m) = mao This shows T is surjective as well, so case n = 0 is proven. Case n > 0: Applying the functor Homc(*,A) to the G-resolution in Lemma 6.1 to get ... <-
HomcCZ[G], A) ~ HomcCZ[G], A) ~ HomcCZ[G], A) ~ Homc(Z, A)
<-
O.
It is known that Homc(Z[G], A) ~ A (see Proposition 8.85 on page 583 of [R]). It follows that
... ~ A ~ A ~ A ~ AC ~ O. By definition of Ext, for n > 0 we have
and
Ext ifGjl(Z, A)
= Kernel (N*)/Image (M*) = Kernel (N)/(g - l)A,
where 9 is a generator of G as in Lemma 6.1. D See also [S], §VIII.4.1 and the Corollary in §VIII.4.
187
(17) If G is a finite cyclic group of order m and A is a trivial G-module then HO(G,A) = A C , H 2n - 1 (G,A) ~ A[m], H2n(G,A) ~ A/rnA,
for all n 2: l. This is a consequence of the previous property.
7. Functorial properties In this section, we investigate some of the ways in which Hn(G, A) depends on G. One way to construct all these in a common framework is to introduce the notion of a "homomorphism of pairs". Let G, H be groups. Let A be a G-module and Ban H-module. If 0: : H ---+ G is a homomorphism of groups and (3 : A ---+ B is a homomorphism of H-modules (using 0: to regard B as an H-module) then we call (0:,(3) a homomorphism of pairs, written
(0:,
(3) : (G, A)
---+
(H, B).
Let G c H be groups and A an H-module (so, by restriction, a Gmodule). We say a map
is transitive if fC2,c3fc 1,c2 = fC 1,c2' for all subgroups G 1 C G 2 C G 3 · Let X* be a G-resolution and X~ a H-resolution, each with a -1 grading. Associated to a homomorphism of groups 0: : H ---+ G is a sequence of Hhomomorphisms
(12)
Theorem 7.1.
(1) If (0:,(3) : (G,A)
---+ (G',A') and (0:',(3') : (G',A') ---+ (G",A") are homomorphisms of pairs then so is (0:' 00:, (3' 0 (3) : (G, A) ---+ (Gil, A").
188
(2) Suppose (a,(J) : (G,A) ---+ (G', A') is homomorphism of pairs, X* is a G-resolution, and X~ is a G'-resolution (each infinite in both directions, with a -1 grading). Let Hn(G, A, X*) denote the derived groups associated to the differential groups Homc(X*, A) with +1 grading. There is a homomorphism (a,f3)x.,X; : Hn(G,A,X*)
---+ Hn(G',A',X~)
satisfying the following properties. (a) IfG = G', A = A', X = X', a = 1 and 13 = 1 then (1, l)x.,x; = 1. (b) If (ex', 13') : (G', A') ---+ (Gil, A") is homomorphism of pairs, X~ is a Gil -resolution then (a'
0
a, 13' of3)x.,x;1 = (a', f3')x;,x;1
(c) If (a,'Y) : (G,A)
---+
0
(a, f3)x.,x;.
(G',A') is homomorphism of pairs then
(a,f3 + 'Y)x.,x; = (a,f3)X.,x;
+ (a, 'Y)x.,x;.
Remark 7.1. For an analogous result for homology, see §§III.8 in Brown
[B]. proof: We sketch the proof, following Weiss, [W], Theorem 2.1.8, pp 52-53. (1): This is "obvious". (2): Let (a,f3) : (G,A) ---+ (G', A') be a homomorphism of pairs. Using (12), we have an associated chain map
a*: Homc(X*, A)
---+ Homcl(X~,A')
of differential groups (Brown §III.8 in [BD. The homomorphism of cohomology groups induced by a* is denoted a~,x.,x; : Hn(G, A, X*) ---+ Hn(G', A', X~).
Properties (a)-(c) follow from §2.2 and the corresponding properties of a*.
o As the cohomology groups are independent of the resolution used, the map (a,f3)x.,x; : Hn(G,A,X*) ---+ Hn(G',A',X~) is sometimes simply denoted by
(13)
189
7.1. Restriction Let X* = X*(G) denote the bar resolution. If H is a subgroup of G then the cycles on G, en (G, A) = HomG(Xn(G), A), can be restricted to H: en(H, A) = HomH(Xn(H), A). The restriction map en(G, A) ---7 en(H, A) leads to a map of cohomology classes:
In this case, the homomorphism of pairs is given by the inclusion map H ---7 G and the identity map f3 : A ---7 A. The map Res is the induced map defined by (13). By the properties of this induced map, we see that Res H,G is transitive: if G c G' C Gil thenj
0: :
Res G',G
0
Res Gil ,G'
= Res G",G.
A particularly nice feature of the restriction map is the following fact. Theorem 7.2. If G is a finite group and G p is a p-8ylow subgroup and if Hn(G, A)p is the p-primary component of Hn(G, A) then (a) there is a canonical isomorphism Hn(G,A) ~ ffipHn(G,A)p, and (b) Res: Hn(G,A) ---7 Hn(Gp,A) restricted to Hn(G,A)p (identified with a subgroup of Hn(G, A) via (a)) is injective. proof: See Weiss, [WJ, Theorem 3.1.15. 0 Example 7.1. Homology is a functor. That is, for any n > 0 and group homomorphism f : G ---7 G' there is an induced homomorphism Hn(f) Hn(G,Z) ---7 Hn(G',Z) satisfying • Hn(gf) = Hn(g)Hn(f) for group homomorphisms f : G G' ---7 Gil, • Hn(f) is the identity homomorphism if f is the identity.
---7
G' 9
The following commands compute H 3 (f) : H 3 (P, Z) ---7 H 3 (85 , Z) for the inclusion f : P <-t 8 5 into the symmetric group 8 5 of its Sylow 2subgroup. They also show that the image of the induced homomorphism H 3 (f) is precisely the Sylow 2-subgroup of H 3 (85 , Z).
r--------------------------- GAP ---------------------------,
I
gap>
S_5:~SymmetricGroup(5);; P:~SylowSubgroup(S_5,2);;
iThere is an analog of the restriction for homology which also satisfies this transitive property (Proposition 9.5 in Brown [BD.
190 gap> gap> gap> gap> gap> gap> gap>
f:~GroupHomomorphismByFunction(P,S_S,
x->x);;
R:~ResolutionFiniteGroup(p,4);; S:~ResolutionFiniteGroup(S_S,4);;
zP_map:~EquivariantChainMap(R,S,f);; map:~TensorWithlntegers(ZP_map);;
Hf:~Homology(map,3);;
Abelianlnvariants(Image(Hf»;
[2,4]
gap>
GroupHomology(S_S,3);
[2,4,3]
e
If H is a subgroup of finite index in then there is an analogous restriction map in group homology (see for example Brown [B], §III.9).
7.2. Inflation Let X* denote the bar resolution of
Xn
e.
Recall
= EB(gl , ... ,gn)EGn R[g1,
... ,gn],
where the sum runs over all ordered n-tuples in e, let xf denote the complex defined by
en. If H
is a subgroup of
X:; = EB(glH, ... ,gnH)E(G/H)nR[g1H, ... ,gnH]. This is a resolution, and we have a chain map defined on n-cells by [g1, ... ,gn] f----+ [g1H, ... ,gnH]. Suppose that H is a normal subgroup of and A is a e-module. We may view AH as a I H-module. In this case, the homomorphism of pairs is given by the quotient map 0: : ----7 H and the inclusion map f3 : A H ----7 A. The inflation map Inf is the induced map defined by (13), denoted
e
e
e
e/
The inflation-restriction sequence in dimension n is
For a proof, see Weiss, [W], §3.4. There an analog of this inflation-restriction sequence for homology. We omit any discussion of transfer and Shapiro's lemma, due to space limitations. Acknowledgements: I thank G. Ellis, M. Mazur and J. Feldvoss, P. Guillot for correspondence which improved the content of these notes.
191
References A.
A. Adem, "Recent devolopments in the cohomology of finite groups," Notices AMS, vol 44(1997)806-812. Available online at http://www.ams.org/notices/199707/199707-toc.html. Bi. M. Bishop, The GAP package CRIME, available from http://www.gap-system.org/Packages/crime.html B. K. Brown, Cohomology of groups, Springer-Verlag, 1982. C. J. Carlson's page: http://www.math.uga.edu/-lvalero/cohointro.html. CEo E. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, 1956. EM. S. Eilenberg, S. MacLane, "Relations between homology and homotopy groups," Proc. Nat. Acad. Sci. U. S. A. 29 (1943). 155-158. E1. G. Ellis, The GAP package HAP 1.8.4, available from http://www.gap-system.org/Packages/hap.html. E2. - - , "Computing group resolutions", J. Symbolic Computation, 28 (2004), 1077-1118. available from http://hamilton.nuigalway.ie/preprints.html. E. L. Evens, The cohomology of groups, Oxford Univ. Press, 1991. Gap. The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.3, 2000 http://www.gap-system.org/. G. R. Godement, Topologie algebrique et theorie des faisceaux, Hermann, 1958. Gr. D. Green's page: http://www.math.uni-wuppertal.de/-green/Coho_v2/. Gu. P. Guillot's page:
http://www-irma.u-strasbg.fr/-guillot/research/cohomology_of_groups/index.html Ho. G. Hochschild, "Relative homological algebra," Trans. Amer. Math. Soc. 82 (1956), 246-269. K. G. Karpilovsky, The Schur multiplier, Oxford Univ. Press, 1987. Ro. M. Roder, The GAP package HAPcryst, available from http://www.gap-system.org/Packages/undep.html. R. J. Rotman, Advanced modern algebra, Prentice Hall, 2002. S. J.-P. Serre, Local fields, Springer-Verlag, 1979. Sh. S. Shatz, Profinite groups, arithmetic, and geometry, Princeton Univ. Press, 1972. St. William Stein, SAGE Mathematics Software (Version 2.9), The SAGE Group, 2007, http://www.sagemath.org. W. E. Weiss, Cohomology of groups, Academic Press, 1969.
Doubles of Residually Solvable Groups Delaram Kahrobaei
Mathematics Department, New York City College of Technology The City University of New York, 300 Jay Street, Brooklyn, NY 11201 CUNY Graduate Center, Doctoral Program in Computer Science 365 Fifth Avenue, New York, NY 10016 E-mail: [email protected] Abstract: In this paper we study residual solvability of doubles of residually solvable groups. We find suitable conditions where this kind of structure is residually solvable, and show that in general this is not the case. However this kind of structure is always meta-residually-solvable. *
Keywords: residually solvable, solvable separability, doubles
1. Introduction and Motivation
The notion of residual properties was first introduced by Philip Hall in 1954 (see [9], page 349). Let X be a class of groups. Gis residually-X if for every non-trivial element 9 in G there is an epimorph of G in X such that the element corresponding to 9 is not the identity. In this paper we consider the question of residual solvability of amalgamated products of residually solvable groups. That is when the free product with amalgamation of two residually solvable groups again residually solvable. The question of being residually solvable can be simplified to whether the group is simple. Neumann in [16] asks the following question: Is it possible that the free product with amalgamation {A * Bj H = K}, where A, B are free groups of finite ranks, H, K are finitely generated subgroups of A,B, respectively, is a simple group? For the case where the amalgamated subgroup is not finitely generated, Ruth Camm [7] constructed an example of a simple free product with amalgamation G = {A * Bj H = K} where A, B are free groups of finite rank and their subgroups H, K have infinite rank. This example can be thought as amalgamated product of two residually *2000 Mathematics Subject Classification. Primary 20E06.
192
193
solvable groups that is not residually solvable. (Le. where the amalgamated subgroup fails the maximal condition). In fact she showed that there exist continuously many non-isomorphic simple amalgamated products of two finitely generated free groups with non-finitely generated amalgamation. For the case where the amalgamated subgroup is finitely generated Burger and Mozes [6] constructed an infinite family of torsion free finitely presented simple groups isomorphic to an amalgamated product {F * F; G} where F, G are finitely generated free groups. Note that a partial answer to the question in [16], is that under the same conditions of the problem, {A * B; H = K} is not simple provided either of indices [A: H], [B : K] is infinite (see [10]). By a double of a group A we mean an amalgamated product G = {A * A; = C} where - is an isomorphism from A onto itself. Clearly, doubles of free groups are never simple. In this paper, we study residual solvability of amalgamated products of finitely generated residually solvable groups. Since free groups are residually solvable, what we prove here is stronger than the corresponding result for the class of free groups. We note that any free product of residually solvable groups is residually solvable again [8]. Here we focus attention on the class of residually solvable groups and show that the doubles of residually solvable groups are residually solvable if we impose the condition of solvable separability over the amalgamating subgroup:
e
Theorem 1.1. Let" - " be an isomorphism from a group A onto itself, be a subgroup of A, and G be the amalgamated product of A and A amalgamating e with C, that is
e
G = {A If A is residually solvable and residually solvable.
e
* A; e = C}. is solvably separable in A then G is also
In general we show that doubles of residually solvable groups are not residually solvable; note that these are one of the only non-trivial (simple or perfect groups) known non-residually solvable groups. However they are meta-residually solvable (see Corollary 3.1). Here is the statement of the Theorem: Theorem 1.2. Let A be a finitely generated residually solvable group, and e a normal subgroup of A, such that A/e is perfect. Let ,,- " be an isomorphic mapping of A onto itself. Then
D
= {A * A; e = C}
194
is meta-residually solvable, but not residually solvable.
Here is an example of Theorem 1.2. Let A be a free group of rank two, there is a natural homomorphism from A onto A5 which is simple. Let be the kernel of such a homomorphism. Clearly Ale is simple (perfect). Forming D = {A * .4; = C} gives the example. Trivially, solvable groups are residually solvable, and it is known that free groups are residually solvable. The author has studied residual solvability of generalized free products of finitely generated nilpotent groups (see [11], [12]), and proved that such structures are often residually solvable. Such amalgamated products occur in one-relator groups (see [4]) and studying residual solvability of generalized free products could be also fruitful in answering old questions on residual solvability of one-relator groups. Studying residual solvability of groups was pioneered by work of Baumslag in 1971, when he proved positive one-relator groups are residual solvable (see [3]). Residual solvability of non-positive groups has been studied in very few cases, for example Kropholler in [13J shows that Baumslag-Solitar groups are free-by-solvable showing that the second derived subgroup is free, therefore it is residually solvable. In [1], Arzhantseva, de la Harpe together with the author define the full prosolvable completion of a group, and these theorems have potential applications to the natural questions that arise.
e
e
Acknowledgment I thank my Ph.D. supervisor G.Baumslag and also K.J.Falconer for helpful comments. The research of this author has been supported by CUNY research foundation at the City College of the City University of New York and New York Group Theory Cooperative.
2. Preliminaries In this section, we recall some facts and prove some lemmas to be used later on.
2.1. Subgroups of amalgamated products We will use a theorem by Hanna Neumann [15] extensively in this paper. With regard to abstract groups, Hanna Neumann showed in the 1950s that, in general, subgroups of amalgamated products are no longer amalgamated products, but generalized free products, indeed she proved the following:
195
let K be a subgroup of G = {A * B; C}, then K is an HNN-extension of a tree product in which the vertex groups are conjugates of subgroups of either A or B and the edge groups are conjugates of subgroups of C. The associated subgroups involved in the HNN-extension are also conjugates of subgroups of C. As a corollary, if K misses the factors A and B (i.e. K n A = {I} = K n B), then K is free; and if K misses the amalgamated subgroup C (i.e. K n C = {l}), then K = I1iE/Xi * F, where the Xi are conjugates of subgroups of A and Band F is free (see [5] for more information) . Later a description was given by the Bass-Serre theory [17], with groups acting on graphs to give some geometric intuition: the fundamental group of a graph of groups generalizes both amalgamated products, HNN-extensions and tree products.
2.2. Some Lemmas Here we prove some lemmas to be used in proving the main results of this paper.
Lemma 2.1. If A is a g'rOUP, C is a subg'rOup of A, c/J is an isomorphic mapping of A onto a g'rOUp B, and D is the amalgamated p'rOduct of A and B amalgamating C with Cc/J, that is
then there is a homomorphism, 1jJ, f'rOm D onto one of the factors, and the kernel of 1jJ, K, is:
K
=
gp(a(ac/J)-lJa E A).
Furthermore this map is injective on each factor.
Proof. Let a be the homomorphism from A onto itself, and f3 be the homomorphism from B onto the inverse of the isomorphic copy of A, i.e. f3 = c/J-l. These homomorphisms can be extended to a homomorphism from D onto A, ([5] page 103, [14]). Since 1jJ(Cab) = Ca(bc/J-l) it follows easily K = ker1jJ = gp(a(ac/J- 1 )Ja E A). By the way that a and f3 are defined, it follows that this homomorphism is one-to-one restricted to either A or B.
o Lemma 2.2. Let A, B, C, D, K and c/J be as in Lemma 2.1 with C a p'rOper subg'rOup of A. Then K is not central in D, in other words, [K,D] i- {1}.
196
Proof. First, note that the center of Dis eD = eA n eB n C. By Lemma 2.1, KnA = {I}, and hence eAnK = {I}. This implies that [K,DJ # {I}.
o Lemma 2.3. Let A, B, C, D, K and ¢ be as in Lemma 2.1. Furthermore, let C be normal in A. Then K commutes with C, i.e. [C, KJ = 1. Proof. Let c E C, and a E A. Since C
(a-1ca)¢ = (a-1¢)(c¢)(a¢) = (a-1¢)c(a¢). Note that a(a¢)-l E K, so
ca(a¢)-l = a(a¢)-lc;
o
that is [C,K] = {I}. Corollary 2.1. Let A, B, C, D, K and ¢ be as above. Then K is free.
Proof. Since the homomorphism 'ljJ is one-to-one restricted to either A or B, then
KnA = {I} = KnA¢. So, by the theorem of Hanna Neumann mentioned in Subsection 2.1, K is 0 free.
2.3. The filtration approach to residual solvability In this section we provide some background for filtration approach which we will use later. A family (A>.I'\ E A) of normal subgroups of A is termed a solvable filtration of A if AI A>. is solvable for every ,\ E A and n>'EA A>. = {I}. We shall say that H is solvably separable in A if n~=l H A>. = H, that is if and only if n~=l A>. c H. Now let H ~ A, then (A>.I'\ E A) is called an H-filtration of H if n>'EA H A>. = H. Let ¢ : H ---t K be an isomorphism between subgroups H of A and K of B. Two equally indexed filtrations (A>. 1,\ E A) and (B>.I'\ E A) of A and B respectively are termed (H, K, ¢)-compatible if (A>.nH)¢ = B>.nK (\f'\ E A). The following Proposition of Baumslag [2] will help us to prove one of the results: let (A>.J'\ E A), (B>.JA E A) be solvable (H, K, ¢)-compatible filtrations of the residually solvable groups A and B respectively. Suppose
197
(A>JA E A) is an H -filtration of A and (B>.IA If, for every A E A, {A/A>.
E
A) is a K-filtration of B.
* B/B>.;HA>./A>. = KB>./B>.},
is residually solvable, then so is G
= {A * B; H = K}.
3. Doubles of residually solvable groups In this section we prove the theorems concerning the doubles of residually solvable groups.
3.1. M eta-residual-solvability Let X be a group property. Then a group G is meta-X if there exist A and Q of property X and a short exact sequence 1 ----+ A ----+ G ----+ Q ----+ 1. Here we prove that in general the amalgamated products of doubles of residually solvable groups are meta-residually-solvable. Proposition 3.1. Let A be a residually solvable group, C be a subgroup of A, and 11- II be an isomorphic mapping of A onto A. Then the generalized free product of A and A amalgamating C with C, G
= {A * A; C = C}
is an extension of a free group by a residually solvable group. Proof. Let <jJ : G ----+ A; then K = ker<jJ = gp(aa-1Ia E A). K is free by the theorem of Hanna Neumann mentioned in subsection 2.1, since AnK = {1} = Anker<jJ.
Therefore G is an extension of a free group by a residually solvable group.
o Corollary 3.1. G is meta-residually-solvable. Proof. Since free groups are residually solvable, then by Proposition 3.1, G is residually solvable-by-residually solvable. That is to say that G IS meta-residually-solvable. o
3.2. Effect of solvably separability on the amalgamated subgroup and residual solvability, Proof of Theorem 1.1 We now prove that if we impose the solvable separability condition on the amalgamated subgroup of doubles of residual solvable groups then the resulting group is residually solvable.
198
Proof. Assuming C is solvably separable in C and A is residually solvable, we want to show that G is residually solvable. That is we must show that for every non-trivial element (1 i=)d E G, there exists a homomorphism, ¢, from G onto a solvable group S, ¢ : G ----) S, such that d¢ i= 1. We consider two cases: Case 1: Let 1 i= d E A. There exists an epimorphism ¢ from G onto A, so that d¢ = d. Since A is residually solvable, there exists .A E N, such that d rf. oAA, where oAA, is the .A-th derived group of A. Now put S = Ajt5A A, a solvable group of derived length at most .A. Note that the canonical homomorphism, (), from A onto S, maps d onto a non-trivial element in S. Now consider the composition of these two epimorphisms, ()o¢, which maps G onto S. The image of d in S is non-trivial: () 0
Case 2: Let 1 i= d
rf.
¢(d)
= d() = do AAi=s1.
A but d E G. Now d can be expressed as follows:
d=albla2b2···anbn (aiEA-C biEA-C).
Since the equally indexed filtrations, {oAAhEN' and {oAAhEN of A and are compatible, we can form GA :
A
Note that G A is residually solvable (by mapping it to one of the factors and noting that the kernel of the map is free). Consider the canonical homomorphism () from G onto GA. Since C is solvably separable in A, i.e.
n
CoAA = C,
AEN .A E N can be so chosen that ai
rf. CoAA, ai rf. CoAA
(for i
= 1,··· ,n).
Hence aloAA b1oAA··· anoAAbn oAAi=c>.1.
This completes the proof of theorem by using Baumslag's Proposition [2], we recalled in Section 2.3. 0 Note also, if G is residually solvable then C is solvably separable in A, since solvable separability is equivalent to n~l H AA C C.
199
3.3. Solvable separability is a sufficient condition for residual solvability; Proof of Theorem 1.2 Note that the condition of solvable separability of the amalgamated subgroup in the factors, in the case of doubles, is necessary. The following theorem shows that the amalgamated product of doubles is not residually solvable where the factors are residually solvable groups. Proof. D is meta-residually solvable by Corollary 3.1. By Lemma 2.1, there exists an epimorphism from D onto A. Let K be the kernel of this epimorphism. Since G is normal in A, by using Lemma 2.3
Now we want to show that D is not residually solvable. We proceed by contradiction. Suppose D is residually solvable. Let d be a non-trivial element in [K, D]. The existence of such an element is guaranteed by Lemma 2.2. Now assume fJ, is a homomorphism of D onto a solvable group S, so that dfJ, =/=8 1. Since fJ, is an epimorphism, GfJ, is a normal subgroup of S, and by
(* ),
If we can show that S = GfJ" then [KfJ" S] = 1, which implies that dfJ, =81, a contradiction. We now need to show that S = G fJ,. We have that D -7> S induces a homomorphism from D/G onto S/GfJ,. Since A/G and A/e each have a perfect subgroup, this induces a homomorphism from A/G to 1. So, S/GfJ, = 1 and hence GfJ, = S. 0 References 1. G.Arzhantseva,P. de la Harpe and D.Kahrobaei The true prosoluble completion of a group: Examples and open problems, Journal of Geometiae Dedicata, Springer Netherlands 124 (2007), 5-26. 2. G.Baumslag, On the Residual Finiteness of Generalized Free Products of Nilpotent Groups, Trans. Amer. Math. Soc 106 (1963), 193-209. 3. G.Baumslag, Positive One-Relator Groups, Trans. Amer. Math. Soc 156 (1971), 165-183. 4. G.Baumslag, A Survey of Groups with a Single Defining Relation, London Math. Soc. Lecture Note Ser. Camb. Univ. Press, Proc. of Groups St Andrews 121 (1986), 30-58. 5. G.Baumslag, Topics in Combinatorial Group Theory, Birkhauser-Verlag, 1993. 6. M.Burger and S, Mazes, Finitely Presented Simple Groups aqnd Products of Trees, C.R. Acad.Sci. Paris Ser. 1 Math. 7 (1997), 747-752.
200 7. R.Camm, Simple Free Products, J. London Math. Soc. 28 (1953), 66-76. 8. K.W. Gruenberg, Residual Properties of Infinite Solvable Groups, Proc. London Math. Soc. 7 (1954), 29-62. 9. P.Hall, The Splitting Properties of Relatively Free Groups, Proc. London Math. Soc. 4. (1978), 343-356. 10. D.Kahrobaei, A Simple Proof of a Theorem of Karrass and Solitar, Contemp. Math. Soc. 372. (2005), 107-108. 11. D.Kahrobaei, On Residual Solvability of Generalized Free Products of Finitely Genertaed Nilpotent Groups,J. Group Theory (accepted) (2007). 12. D.Kahrobaei, Residual Solvability of Generalized Free Products, PhD Thesis, CUNY Graduate Center (1994) 13. P.H. Kropholler, Baumslag-Solitar Groups and Some Other Groups of Cohomological Dimension Two, Comment. Math. Relv. 65 (1990), 547-558. 14. B.H. Neumann, An Essay on Free Products of Groups with Amalgamation, Philos. Trans. Roy. Soc. London Ser. A 246 (1954), 503-554. 15. H. Neumann, Generalized Free Products with Amalgamated Subgroups. ii, Amer. J. Math. 71 (1949), 491-540. 16. P. Neumann, SQ-universality of Some Finitely Presented Groups, J. Aust. Math. Soc. (1973). 17. J.P. Serre, Trees, Springer-Verlag, (1980).
AN APPLICATION OF A GROUP OF OL'SHANSKII TO A QUESTION OF FINE ET. AL. Seymour Lipschutz
Department of Mathematics, Temple University, Philadelphia, Pennsylvania, 19122 Dennis Spellman
Department of Mathematics, Temple University, Philadelphia, Pennsylvania, 19122
Dedicated to Tony Gaglione on his Sixtieth birthday.
1. Introduction
The classical commensurator of a subgroup may be found in [KM] where it is called the virtual normalizer. Proposition 2.4 of [BBFGRS] asserts that any nontrivial finitely generated subgroup H -=I=- {I} in a free group F has finite index in its commensurator. Following the above proposition the authors of [BBFGRS] define a group G satisfying the commensurator condition as one for which every finitely subgroup H -=I=- {I} has finite index in its commensurator. The authors of [BBFGRS] asked for a finitely generated infinite group G with the commensurator property that is not hyperbolic. In [0] Ol'shanskii constructed a 2-generator infinite simple group with the property that every nontrivial proper subgroup is infinite cyclic. We show that any such group satisfies the commensurator condition but is not hyperbolic. In [BBFGRS] the authors define for finitely generated groups a condition called property R. We give a definition, which coincides with theirs in the case of finite generation, but which applies to arbitrary groups. We consider operators which preserve property R as well as those that do not; moreover, we reflect upon the relationships (or lack thereof) between property Rand various other finiteness conditions. Finally we introduce a generalization of property R which we call weak property R or property WR. The paper is divided into six sections beginning with this Introduction (Section 1). In Section 2 we introduce the commensurator condition. In Section 3 we recall 201
202
various conditions on maximal abelian subgroups and apply these to argue that Ol'shanskii's group is an example of the type we seek. In Section 4 we study property R. In Section 5 we introduce property WR. In Section 6 we pose some questions.
2. The Commensurator Condition In this section we introduce the commensurator of a subgroup and the commensurator condition.
Definition 2.1. Let G be a group and H a subgroup in G. The commensurator of H in G, denoted by commc(H), consists of the elements 9 E G such that
IH: H n H91 <
00
and IH9 : H n H 91 < oo}.
One can easily prove:
Proposition 2.1 (BBFGRS). : If H is a subgroup in the group G, then commc(H) is a subgroup in G. Clearly H is a subgroup of commc(H).
Proposition 2.2 (BBFGRS). If F is a free group then, for every nontrivial finitely generated subgroup H =I- {ll in F, one has [commF(H) :
H] <
00.
Now we define the commensurator condition.
Definition 2.2 (BBFGRS). A group G satisfies the commensurator condition provided, for every nontrivial finitely generated subgroup H =I{l} G one has [commc(H) : H] < 00. Thus free groups satisfy the commensurator condition. The reason this rather technical condition is of some interest is that it implies a finiteness condition, Stalling's Property S, that has been studied in [BBFGRS].
3. Conditions on Maximal Abelian Subgroups and the Counterexample If 9 is an element of the group G let Gc(g) = {x E G : gx = xg} be its centralizer in G. In this section we state some definitions, which we deem sufficiently classical to omit references to the literature.
203
Proposition 3.1. Let G be a group. The following three conditions are pairwise equivalent. (1) The relation commutes with is transitive on G \ {1}. (2) Gc(g) is abelian for every 9 =I- 1 in G (3) For every pair M1 =I- M2 of distinct maximal abelian subgroups in G one has M1 n M2 = {1}. Definition 3.1. A group satisfying anyone (and therefore all three) ofthe conditions of Proposition 3.1 is commutative transitive or CT. Definition 3.2. A subgroup M in a group G is malnormal in G provided 9 E G \ M implies g-l M 9 n M = {1}. Definition 3.3. A group G such that all its maximal abelian subgroups are malnormal is said to be conjugately separated abelian or CSA . Remark 3.1. Every abelian group is trivially both CT and CSA. Proposition 3.2. Every GSA group is GT. Proof. Suppose G is CSA. Let M1 and M2 be distinct maximal abelian subgroups and assume to deduce a contradiction that 9 =I- 1 lies in M 1 nM2 . Let m E M 1 . Then g = m-1gm lies in m- 1 M 2m n M 2. Since Gis CSA this implies m E M 2 . Therefore, since m E M1 was arbitrary, M1 is a subgroup of M2. By maximality M1 = M 2, contrary to hypothesis. Thus, M1 =I- M2 implies M1 n M2 = {1} whenever M1 and M2 are maximal abelian in G. Hence, G is CT whenever it is CSA. 0 Remark 3.2. The infinite dihedral group D b- 1 > is CT but not CSA.
=< a, b; a 2
To see this note that a free product of abelian groups is CT. On the other a CSA group cannot contain a nontrivial abelian subgroup. However in D the commutator subgroup is cyclic and hence D cannot be CSA. Proposition 3.3. (Myasnikov and Remeslennikov): Let G be a group such that Gc(g) is infinite cyclic for all 9 =I- 1 in G. Then G is GSA. Proof. If G = {1} then the result is trivially true. Now assume that G =I{1} satisfies the hypotheses of the proposition and observe that G must be torsion free and CT. Let M =< m > be maximal abelian in G where, m =I- 1 and is not a proper power in G. Suppose to deduce a contradiction
204 that 9 E G \ M but g-1Mg n M i= {I}. Then 9 does not commute with m. In particular, 9 i= 1. Since G is CT and the maximal abelian subgroups M and g-1 Mg intersect nontrivially we must have g-1 M 9 = M and the inner automorphism x ---> g-1xg of G restricts to an automorphism of M. Hence, g-1mg = m or g-1mg = m- 1 . But we are assuming that 9 does not commute with m so, g-1mg = m- 1. Then g-2mg 2 = g-1m- 1g = m and g2 commutes with m. g2 i= 1 since G is torsion free. But 9 commutes with g2 ,which commutes with m, so 9 commutes with m since G is CT. This contradicts 9 1:. M = Cc(m). The contradiction shows that the condition g-1 Mg n M i= {I} implies gEM. Thus G is CSA. 0 In [BBFGRS] it was asked to give an example of finitely generated group that is neither free nor hyperbolic but which satisfies the commensurator condition. We show that a group constructed by Ol'Shanskii provides such an example. In [OJ Ol'shanskii constructed a 2-generator infinite simple group with the property that every proper nontrivial subgroup is infinite cyclic. For the remainder of this section let G be one fixed such group, which shall be called Olshanskiis group. Since G is a nonabelian simple group it is centerless. Thus, for each 9 i= 1 in G the centralizer Cc(g) is proper in G (and nontrivial since 9 E Cc(g)) hence infinite cyclic. It follows from Proposition 3.1 that G is CSA. We claim that Proposition 3.4. The group G satisfies the commensurator condition and further G is not hyperbolic. Proof. We first show that G satisfies the commensurator condition. In order to do this we must demonstrate that, if H i= {I} is a finitely generated subgroup in G, then [commc(H) : HJ < 00. First of all observe that every subgroup of G is finitely generated so that the finiteness condition is redundant. Note also that commc(G) = G and thus [commc(G) : G] = 1 < 00. So we must show, for every proper subgroup H i= {I} in G, [commc(H) : H] < 00. Now if commc(H) were infinite cyclic wed be finished since every nontrivial subgroup of an infinite cyclic group has finite index in the containing group. In other words it suffices to show that commc(H) is a proper subgroup of G whenever H i= {I} is proper in G. Now if H i= {I} is proper in G it is cyclic so H extends to a maximal abelian subgroup M. Let 9 E G \ M. We claim that 9 1:. commc(H). Suppose to deduce a contradiction that 9 E commc(H). Then, in particular, [H : g-1 H g7 H] < 00. This implies that g-1Hg n H i= {I} since H is infinite cyclic. But then g-1Mg n M
205 g-l H 9 n H -=I=- {1} and so gEM since G is eSA. That contradicts the choice of g. The contradiction shows 9 tf. commc(H). Hence, commc(H) is proper in G whenever H -=I=- {1} is proper in G and we are finished. This shows that G satisfies the commensurator condition. 0
To show that G is not hyperbolic we need the following definition.
Definition 3.4. (Fine and Rosenberger [FR]) A group H is restricted Gromov or RG provided, if 9 and h are any elements in H, then either the subgroup < g, h > is cyclic or there exists a positive integer t with gt -=I=- 1, ht -=I=- 1 and < gt, ht > is the free product < gt > * < ht >. We state without proof
Proposition 3.5. (Gromov): Torsion free hyperbolic groups are RG. Proof. We can deduce that G is not hyperbolic by showing that Olshanskiis group G is not RG. Suppose that a and b generate the 2-generator group G. If G were RG there would exist a positive integer t such that < at, bt >=< at > * < bt >. Now < at > and < bt > are each infinite cyclic since G is torsion free. Thus, < at, bt > is free of rank 2. Since every proper subgroup of G is cyclic we conclude G =< at, bt > is free of rank 2 an obvious contradiction. The contradiction shows that G is not hyperbolic completing the proposition. 0 4. Property R Definition 4.1. A group G satisfies Property R provided, for every finite subset X ~ G and every subgroup H ~< X >, it is the case that H has finite index in < X > if and only if, for each x EX, there is a positive integer n(x) such that xn(x) E H. Definition 4.2. A group G satisfies Stallings Property S if for any finitely generated subgroups A and B of G with [A : AnB] < 00 and [B : AnB] < 00 we have [< A, B >: An B] < 00, that is for any two commensurable finitely generated subgroups in G the intersection must have finite index in the join. Definition 4.1 coincides with that given in [BBFGRS] in the case G is finitely generated the only case for which property R was defined in [BBFGRS]. Property R is of interest because it implies property S and because
206 there are many examples. Every nilpotent group satisfies property R (Malcev [M]) as does every direct product G x F where G satisfies property R and F is finite [BBFGRSj. The question was asked whether there exists a group G satsifying Property R but not virtually nilpotent. Recall that if P is a group property then G is virtually-P if G has a subgroup of finite index which satisfies P. In [BBFGRS] the following example was given which answered the question.
Example 4.1 (BBFGRS). : Let n ~ 2 be an integer. Then the group En =< a, t; rlat = an > satisfies properly R but is not virlually nilpotent. The following proposition provides another example of such a group. Proposition 4.1. Ol'Shanskii's group G satisfies Properly R and is not virlually nilpotent, Proof. Let X be a finite subset of Olshanskiis group G. If < X > is cyclic then it is nilpotent so satisfies property R. Thus to show that G satisfies property R it suffices to take a finite subset X = Xl" Xk which generates G. We may assume without loss of generality that 1 tt X. Now suppose that H is a subgroup of infinite index in G. Thus, H is a proper subgroup of G. Now if x;(j) E H where n(j) is a positive integer, j = 1" k, then
the x~(j) commute in pairs since H, being proper, is cyclic. Moreover, since G is torsion free and no Xj = 1, we must have x;(j) -:j 1, j = 1" k. By commutative transitivity the Xj must commute in pairs. But, since the Xj generate G, that would imply that G is abelian a contradiction. The contradiction shows that G satisfies property R. It is an easy exercise to show that the only subgroup of finite index in an infinite simple group is the entire group. No nonabelian simple group can be nilpotent. Thus, Ol'shanskii's group is another example of a property R group which is not virtually nilpotent. 0
Example 4.2. Let A =< a, b; ab = ba > be free abelian of rank 2 with basis {a, b}. A certainly satisfies property R. Let T E Aut( A) be determined by transposing a and b. Let G
=< a,b,t;ab = ba,Clat = b,Clbt = a,t 2 = 1 >
be the semidirect product determined from this data. Let X = {a, t}. Since b = rlat, X generates G. Let H =< a >. Then H has infinite index in G =< X >. Indeed, H =< a > has infinite index in the subgroup
207
A =< a, b >. But a l R.
=a
E
Hand t 2
=
1 E H. Thus G violates property
The authors of [BBFGRS] asked for a characterization of those finite extensions of property R groups which retain property R. We believe that, in some sense, the direct product result is best possible since the semidirect product of Example 4.2 violates property R.
Proposition 4.2. Property R is preserved in subgroups, homomorphic images and direct limits. Proof. It is obvious from our definition of property R that it is inherited by subgroups. Suppose next that G satisfies property Rand ¢ : G ----> G' is an epimorphism. Let Y = {YI" yd be a finite subset of G' and suppose H is a subgroup of < Y >. Suppose further that there are positive integers n(j) such that Y7(j) E H, j = 1" k. We want to show that H has finite index in < Y >. For each 1 :<:; j :<:; k fix a preimage Xj E G of Yj under ¢. We claim that U(YI"Yk)V(YI"Yk)-1 ~
H
implies that
If not then
U(YI"Yk)V(YI"Yk)-1 E< y~(l)"y~(k) >c H. Thus, if there were infinitely many distinct right cosets of H in < Y >, there would be infinitely many distinct right cosets of < X~(l)" x~(k) > in < Xl, ,Xk >. This contradicts the fact that G satisfies property R. The contradiction shows that [< Y >: H] < 00. Hence, G' satisfies property R and property R is preserved under homomorphic images. Finally suppose that A is an upward directed set and (G.x).xEA is a family of property R groups indexed by A. Suppose that, for each ,J1, E A with >. < J1, there is a homomorphism P.x,!, : G.x ----> G and suppose further that, for each >., J1" 1/ E A with>' < J1, < 1/, one has p.x,v = P.x,!'P!',v where we write our maps to the right of their arguments and compose accordingly. Let G be the direct limit determined from this data and, for each>' E A, let P.x : G.x ----> G be the limit map. It will suffice to show that, if r = hI, ... , ,d is any finite subset of G, then the subgroup of G generated by r satisfies property R. Fix such a subset r. Then there is >. E A and gl, ... ,gk E G.x
208 such that "/j = gjP>', j = 1" k. Now < "/1, ... , "/k > is a subgroup of the homomorphic image G>.p>.. Hence, < "/1, ... , "/k > satisfies property R. 0 Let G be a finitely generated group. We wish to examine the relationship between property R and various other finiteness conditions for G. Specifically we consider (1) The maximum condition. (2) The minimum condition. (3) Residual finiteness. (4) Hopficity. The following will be useful for our purposes. Lemma 4.1. Let G be an infinite group generated by finitely many torsion elements. Then G does not satisfy property R.
Proof. Suppose X
= {Xl, ... , xd
n(j) 2: 2,j = 1, ... , k. Then H H,j = 1, ... ,k.
generates G and x?(j)
= {I}
=
1 where each
has infinite index in G yet x?(j) E 0
For example, the infinite dihedral group presented as D =< a, b; a 2 = 1, a-lba = b- l > is generated by {a, ab} and each of a and ab has order 2 in D. Thus D violates property R. In the same paper in which Ol'shanskii constructed the group in Section 3 he showed how to modify the construction to create a 2-generator infinite group in which every proper subgroup is finite cyclic. Since such a group clearly satisfies both the maximum and minimum conditions neither one (nor both) implies property R as a consequence of Lemma 4.1 applied to Ol'shanskii's modified construction. Conversely property R implies neither condition. The groups B N of Example 4.1 violate the maximum condition as the normal closure of < a > is isomorphic to the additive group of the ring Z[~l and is not finitely generated as a group. The infinite cyclic group, for example, satisfies property R but violates the minimum condition. The infinite dihedral group D =< a, b; a 2 = 1, a-1ba = b- 1> is easily seen to residually be in the family of finite dihedral groups Dn =< a, b; a 2 = 1, bn = 1, a-1bna = bn - 1 > of order 2n. Hence, D is a finitely generated residually finite group and thus also Hopfian. This shows that the Hopf property or even residual finiteness does not imply property R. On the other hand Ol'shanskii's group of Section 3 is not residually finite as the only finite image is the trivial group {I}. Ol'shanskii's group is, however, Hopfian as it is easy to see every simple group must be.
209 Proposition 4.3. Property R is not preserved in unrestricted direct products or even unrestricted direct powers. Proof. Let N be the set of positive integers. Since the infinite dihedral group D is residually in the family {Dn : n E N} of finite dihedral groups, D embeds in the unrestricted direct product IInENDn. If IInENDn satisfied property R then so would the subgroup D a contradiction. Each Dn, being finite, satisfies property R. Let Soo, the infinite symmetric group, be the group of all permutations of N which move only finitely many integers. For each n E N fix an embedding of Dn into Soo. Now Soo satisfies property R since it is the direct limit of the family {Sn : n E N} of finite symmetric groups. But IInENDn embeds in the unrestricted direct power S!. Hence, S! violates property R. D 5. Weak Property R Proposition 5.1. The following conditions on a group G are equivalent. (WR1) If Go is any subgroup of G and Go* is any homomorphic image of Go, then the set of torsion elements in G o* forms a locally finite subgroup of G o*. (WR2) If X is any finite subset of G and N is any normal subgroup of < X >, then N has finite index in < X > if and only if for each x E X there is a positive integer n(x) such that xn(x) EN. Note that the set of torsion elements in a property S group forms a locally finite subgroup. However, property S is not in general preserved in homomorphic images. This latter fact can be seen from the result that free groups satisfy property S. Proof. (of Proposition 5.1) WR1
===}
WR2
Assume G satisfies WRl. Let X = {Xl, ... , xd be a finite subset of G and let Go =< X >. Let N be a normal subgroup of Go. Suppose that for j = 1, ... , k there is a positive integer n(j) such that x;(j) E N. Then {N Xl, ... , N Xk is contained in the set T of torsion elements of the homomorphic image GaiN of Go. Thus, GaiN =< NXI, ... , NXk > is a finite group and so [< X >: N] = [Go: N] < 00. Hence G satisfies WR2. WR2
===}
WR1
210
Suppose G satisfies WR2. Let Go be a subgroup of G and let ¢ : Go - ? G o* be an epimorphism. It will suffice to show that, if T is the set of torsion elements in G o* and (91" 9k) E Tk is a finite tuple of elements ofT, then the subgroup < 91" 9k > of G o* is finite. For each j = 1" k choose a preimage Xj E Go of 9j under ¢. Let H =< X1"Xk > nKer(¢). Now since each Xj maps into T there is a positive integer n(j) such that x7(j) E H, j = 1, ,k. Furthermore, letting X = X1"Xk, H is normal in < X > since Ker(¢) is normal in Go and H =< X > nKer(¢). Since G satisfies WR2 we must have that H has finite index in < X >. Then
< X > j H = < X > j ( < X > nK er( ¢)) = « X> Ker(¢)jKer(¢) =< X1¢"Xk¢ >=< gl"gk > is finite. Hence G satisfies WRl.
o
Definition 5.1. A group satisfies weak property R or property WR provided it satisfies either one (and therefore both) of the conditions WR1 or WR2 of Proposition 5.1. Definition 5.2. A group is a U-group unique.
if roots, when they exist, are
Definition 5.3. A group is an FC-group provided every conjugacy class is finite. Proposition 5.2. (B.H. Neumann [N}): A torsion free FC-group is abelian. Proposition 5.3. Let G be a torsion free property WR group. Then Gis aU-group. We note that both property WR1 and the fact that torsion freeness implies uniqueness of roots within the category are well-known properties of nilpotent groups. Proof. (of Proposition 5.3) Suppose that G is a torsion free property WR group. Suppose n is a positive integer and x, y E G satisfy xn = yn. We must show that x = y. We may assume without loss of generality that G =< x, Y > since property WR is clearly inherited by subgroups. Let xn = Z = yn. Then z is central in G so < z > is normal in G. Moreover, since xn = yn = z and {x,y} generates G we must have [G :< z >J < 00 as G satisfies property WR. Now let w E G be arbitrary. Then
211
< z ><:;;; Gc(w) and so from [G :< z >] = [G : Gc(w)][Gc(w) :< z >] we see that [G: Gc(w)] < 00. It follows that G is an Fe-group. By Proposition 5.2, G is abelian. But then, in the torsion free abelian group G, xn = yn implies (xy-l)n = 1 which in turn implies x = y. 0 We note that a slight variation of the proof shows that torsion free property S groups are U-groups. 6. Questions Question 6.1. Must every finitely generated property R group be Hopfian? Question 6.2. Are property R groups closed under finite direct products? Are they closed under finite direct powers? Question 6.3. Does property WR imply property R? Question 6.4. Must every torsion free property WR group embed in a property WR group admitting roots? What about torsion free property R groups? What about torsion free property S groups? 7. References [BBFGRS] G. Baumslag, O. Bogopulski, B. Fine, A.M. Gaglione, G. Rosenberger and D. Spellman, On some finiteness properties in infinite groups Alg. Colloquium, 15(1), 2007, 1-22 [FR] B. Fine and G. Rosenberger, On restricted Gromov groups Comm.
In Alg., 20(8), 1992, 2171 2181 [KM] I. Kapovich and A.G. Myasnikov, Stallings foldings J. Alg. , 248, 2002, 608 668 [M] A.I. Malcev, Nilpotent torsion free groups Izv. Akad. Nauk. SSSR, 67, 1949, 347 366 [N] B.H. Neumann, Groups with finite classes of conjugate elements Proc. London Math. Soc., 3, 1951, 178 187
[0] A.Yu. Olshanskii, Infinite groups with cyclic subgroups Soviet Math. Dokl., 20(2), 1979,343 346
Quotient Isomorphism Invariants of a Finitely Generated Coxeter Group Michael Mihalik
Mathematics Department, Vanderbilt University, Nashville TN 37240, USA John Ratcliffe
Mathematics Department, Vanderbilt University, Nashville TN 37240, USA Steven Tschantzk
Mathematics Department, Vanderbilt University, Nashville TN 37240, USA
1. Introduction
The isomorphism problem for finitely generated Coxeter groups is the problem of deciding if two finite Coxeter matrices define isomorphic Coxeter groups. Coxeter [4]] solved this problem for finite irreducible Coxeter groups. Recently there has been considerable interest and activity on the isomorphism problem for arbitrary finitely generated Coxeter groups. In this paper we describe a family of isomorphism invariants of a finitely generated Coxeter group W. Each of these invariants is the isomorphism type of a quotient group WIN of W by a characteristic subgroup N. The virtue of these invariants is that WIN is also a Coxeter group. For some of these invariants, the isomorphism problem of WIN is solved and so we obtain isomorphism invariants that can be effectively used to distinguish isomorphism types of finitely generated Coxeter groups. We emphasize that even if the isomorphism problem for finitely generated Coxeter groups is eventually solved, several of the algorithms described in our paper will still be useful because they are computationally fast and would most likely be incorporated into an efficient computer program that determines if two finite rank Coxeter systems have isomorphic groups. 212
213
In §2, we establish notation. In §3, we describe two elementary quotienting operations on a Coxeter system that yields another Coxeter system. In §4, we describe the binary isomorphism invariant of a finitely generated Coxeter group. In §5, we review some matching theorems. In §6, we describe the even isomorphism invariant of a finitely generated Coxeter group. In §7, we define basic characteristic subgroups of a finitely generated Coxeter group. In §8, we describe the spherical rank two isomorphism invariant of a finitely generated Coxeter group. In §9, we make some concluding remarks.
2. Preliminaries A Coxeter matrix is a symmetric matrix M = (m(s, t))s,tES with m(s, t) either a positive integer or infinity and m(s, t) = 1 if and only if s = t. A Coxeter system with Coxeter matrix M = (m(s, t))s,tES is a pair (W,5) consisting of a group Wand a set of generators 5 for W such that W has the presentation W
=
(5 I (st)m(s,t) : s, t E 5 and m(s, t) < 00).
We call the above presentation of W, the Coxeter presentation of (W, 5). If (W,5) is a Coxeter system with Coxeter matrix M = (m(s, t))s,tES, then the order of st is m(s, t) for each s, tin 5 by Prop. 4, p. 92 of Bourbaki [3], and so a Coxeter system (W,5) determines its Coxeter matrix; moreover, any Coxeter matrix M = (m(s, t))s,tES determines a Coxeter system (W, 5) where W is defined by the corresponding Coxeter presentation. If (W, 5) is a Coxeter system, then W is called a Coxeter group and 5 is called a set of Coxeter generators for W, and the cardinality of 5 is called the rank of (W, 5). A Coxeter system (W,5) has finite rank if and only if W is finitely generated by Theorem 2 (iii), p. 20 of Bourbaki [3]. Let (W,5) be a Coxeter system. A visible subgroup of (W, 5) is a subgroup of W of the form (A) for some A <;;; 5. If (A) is a visible subgroup of (W,5), then ((A),A) is also a Coxeter system by Theorem 2 (i), p. 20 of Bourbaki [3]. When studying a Coxeter system (W,5) with Coxeter matrix M it is helpful to have a visual representation of (W,5). There are two graphical ways of representing (W,5) and we shall use both depending on our needs. The Coxeter diagram (C-diagram) of (W,5) is the labeled undirected graph ~ = ~(W, 5) with vertices 5 and edges {(s,t): s,t E 5 and m(s,t) > 2} such that an edge (s, t) is labeled by m(s, t). Coxeter diagrams are useful
214
for visually representing finite Coxeter groups. If A c S, then ,6.( (A), A) is the sub diagram of ,6.(W, S) induced by A. A Coxeter system (W, S) is said to be irreducible if its C-diagram ,6. is connected. A visible subgroup (A) of (W, S) is said to be irreducible if «(A), A) is irreducible. A subset A of S is said to be irreducible if (A) is irreducible. A subset A of S is said to be a component of S if A is a maximal irreducible subset of S or equivalently if ,6.( (A), A) is a connected component of ,6.(W, S). The connected components of the ,6.(W, S) represent the factors of a direct product decomposition of W. The presentation diagram (P-diagram) of (W, S) is the labeled undirected graph r = r(W, S) with vertices S and edges {(s,t): s,t E Sand m(s,t) < oo}
such that an edge (s, t) is labeled by m(s, t). Presentation diagrams are useful for visually representing infinite Coxeter groups. If A c S, then r( (A), A) is the sub diagram of r(W, S) induced by A. The connected components of r(W, S) represent the factors of a free product decomposition of W. For example, consider the Coxeter group W generated by the four reflections in the sides of a rectangle in E2. The C-diagram of (W, S) is the disjoint union of two edges labeled by 00 while the P-diagram of W is a square with edge labels 2. Let (W, S) and (W', S') be Coxeter systems with P-diagrams rand r ' , respectively. An isomorphism ¢ : (W, S) ----> (W', S') of Coxeter systems is Hn isomorphism ¢ : W ----> W' such that ¢(S) = S'. An isomorphism 'Ij; : r ----> r ' of P-diagrams is a bijection from S to S' that preserves edges and their labels. Note that (W, S) ~ (W', S') if and only if r ~ r'. We shall use Coxeter's notation on p. 297 of [5 for the irreducible spherical Coxeter simplex reflection groups except that we denote the dihedral group D~ by D2(k). Subscripts denote the rank of a Coxeter system in Coxeter's notation. Coxeter's notation partly agrees with but differs from Bourbaki's notation on p.193 of [3]. Coxeter [4]] proved that every finite irreducible Coxeter system is isomorphic to exactly one of the Coxeter systems An, n ~ 1, B n , n ~ 4, Cn, n ~ 2, D 2 (k), k ~ 5, E 6 , E 7 , E s , F 4 , G 3 , G 4 . For notational convenience, we define B3 = A 3 , D2(3) = A 2, and D 2(4) = C 2 The type of a finite irreducible Coxeter system (W, S) is the isomorphism type of (W, S) represented by one of the systems An, B n , Cn, D 2 (k), E 6 ,
215
E 7 , E s , F4, G 3 , G 4 · The type of an irreducible subset A of S is the type of ((A),A). The C-diagram of An is a linear diagram with n vertices and all edge labels 3. The C-diagram of Bn is a V-shaped diagram with n vertices and all edge labels 3 and two short arms of consisting of single edges. The Cdiagram of is a linear diagram with n vertices and all edge labels 3 except for the last edge labelled 4. The C-diagram of D2 (k) is a single edge with label k. The C-diagrams of E 6 , E 7 , Es are star shaped with three arms and all edge labels 3. One arm has length one and another has length two. The C-diagram of F 4 is a linear diagram with edge labels 3,4, 3 in that order. The C-diagram of G 3 is a linear diagram with edge labels 3,5. The C-diagram of G 4 is a linear diagram with edge labels 3,3,5 in that order.
en
3. Elementary Quotient Operations In this section we describe two types of elementary edge quotient operations on a Coxeter system (W, S) of finite rank. The first we call edge label reduction and the second we call edge elimination. Suppose sand t are distinct elements of S with m(s, t) < 00. Let d be a positive divisor of m = m(s, t), with d < m, and let N be the normal closure of the element (st)d of W. Then a presentation for WIN is obtained from the Coxeter presentation for (W, S) by adding the relator (st)d. As m = (mld)d, the relator (st)m is derivable from the relator (st)d and so the relator (st)m can be removed from the presentation for WIN. Assume d > 1. Then the presentation for WIN is a Coxeter presentation whose P-diagram is obtained from the P-diagram for (W, S) by replacing the label m on the edge (s, t) with the label d. We call the operation of passing from the Coxeter system (w, S) to the quotient Coxeter system (WIN,{sN: s E S}) the (s,t) edge label reduction from m to d. For example, if we reduce the 4 edge of F 4 to 2, we obtain the Coxeter system A2 x A 2. Now assume d = 1. We delete from the presentation for WIN the generator t and the relator st and replace all occurrences of t in the remaining relators by s. Suppose r is in Sand k = mer, s) < 00 and e = mer, t) < 00. Then we have the relators (rs)k and (rs)l in the presentation for WIN. Let d be the greatest common divisor of k and e. Then there are integers a and b such that d = ak+be. This implies that (rs)d is derivable from (rs)k and (rs)l and so we may add the relator (rs)d to the presentation for WIN. Then (rs)k and (rs)l are derivable from (rs)d and so we can eliminate the relators (rs)k and (rs)l from the presentation for WIN. We do this for each
216
r in 8 such that m(r, s) <
00 and m(r, t) < 00. On the P-diagram level, we have eliminated the edge (s, t) and identified the vertices sand t and we have coalesced each edge (r, s) with label k < 00 with the edge (r, t) with label £ < 00 to form an edge with label d the greatest common divisor of k and £. If each common divisor d is greater than one, we obtain a Coxeter presentation for WIN. If some common divisor d is one, we delete the corresponding generator r and repeating the above reduction procedure on the presentation of WIN. As the set 8 of generators is finite, we will eventually stop deleting generators and obtain a Coxeter presentation for WIN with generators the subset 8' of {sN : s E 8} corresponding to the undeleted elements of 8. We call the operation of passing from the Coxeter system (W, 8) to the quotient Coxeter system (WIN, 8') the (s, t) edge elimination. For example, if we eliminate the 3 edge from C 3 , we obtain the Coxeter system Al x AI'
4. The Binary Isomorphism Invariant Let (W,8) be a Coxeter system of finite rank. For each pair of elements s, t of 8 with m(s, t) < 00, let b(s, t) be the 2-part of m(s, t), that is, b(s, t) is the largest power of 2 that divides m(s, t). If m(s, t) = 00, we define b(s, t) = 00. Let Nb be the normal closure in W of all the elements of the form (st)b(s,t) with b(s, t) < 00, and let Wb = WINb. Let TJ : W --> Wb be the quotient homomorphism, and let 8 b = TJ(8).
Proposition 4.1. The pair (Wb,8b) is a Coxeter system such that if s
and t are in 8, then TJ(s) = TJ(t) if and only if sand t are conjugate in W. If sand tare nonconjugate elements of 8, then the order of TJ( S )TJ( t) is the minimum of the set of all b( u, v) such that u and v are in 8 and u is conjugate to s and v is conjugate to t. In particular, the order of TJ(s)TJ(t) is a power of 2 or 00. Proof. The system (Wb,8b) can be obtained from (W,8) by a sequence of elementary quotient operations. First we can do a series of edge label reductions of all the even labelled edges of the P-diagram of (W,8) to their 2-parts. Then we do a series of edge eliminations of all the odd labelled edges. Each element of the form (st)b(s,t) with b(s, t) < 00 is in the commutator subgroup of W. Therefore abelianizing W factors through the quotient WINb, and so TJ(s) = TJ(t) if and only if sand t are the same odd component of the P-diagram of (W, 8). Hence TJ(s) = TJ(t) if and only if s and t are conjugate in W by Prop. 3, p. 12 of Bourbaki [3].
217
Suppose sand tare nonconjugate elements of Sand u and v are in S, with m( u, v) < 00, and u is conjugate to s and v is conjugate to t. Then u and v are not conjugate, and so m( u, v) is even, and therefore b( u, v) is a power of 2 greater than 1. In the coalescence of two such edges, the greatest common divisor is the minimum of the two edge labels. Therefore the order of TJ( s )TJ( t) is the minimum of the set of all b( u, v) such that u and v are in Sand u is conjugate to s and v is conjugate to t. 0
Theorem 4.1. Let (W, S) be a Coxeter system oj finite rank. For each pair oj elements s, t oj S with m(s, t) < 00, let b(s, t) be the largest power oj 2 that divides m(s, t). Let Nb be the normal closure in W oj all the elements oj the Jorm (st)b(s,t) with m(s, t) < 00, Then Nb is the normal closure in W oj the set oj all elements oj W oj odd order. ThereJore Nb is a characteristic subgroup oj W that does not depend on the choice oj Coxeter generators S.
Proof. Every element of the form (st)b(s,t) with m(s, t) < 00 has odd order, and so Nb is contained in the normal closure of all the elements of odd order. Let w be an element of odd order, then TJ(w) has odd order in Wb = W/Nb. By the previous proposition, (Wb, Sb) is a Coxeter system with all edge labels a power of 2. Therefore TJ(w) is conjugate to an element of odd order of a finite visible subgroup of (Wb, Sb) by [3], Ch. V, §4, Ex. 2. The finite visible subgroups of (Wb, Sb) are direct products of groups of type Ai and C 2 , and so are 2-groups. Therefore Wb has no nontrivial elements of odd order. Hence TJ(w) = 1, and so w is in Nb. Thus Nb is the normal closure of all the elements of W of odd order. 0
P. Bahls proved in his Ph.D. thesis [1] that any finitely generated Coxeter group has at most one P-diagram, up to isomorphism, with all edge labels even; see Theorem 5.2 in Bahls and Mihalik [2]. Therefore the isomorphism type of the P-diagram of (Wb, Sb) is an isomorphism invariant of W by Theorem 4.2. We call the isomorphism type of the P-diagram of (Wb, Sb) the binary isomorphism invariant of W. In Figure 1, we illustrate two P-diagrams and their binary isomorphism invariant P-diagrams below them. The even diagrams are not isomorphic, and so the top two P-diagrams represent nonisomorphic Coxeter groups.
218
3
4 4
4
3
4
2
~4
~2
2
4
Figure 1
5. Matching Theorems Let (W, 8) be a Coxeter system. A basic subset of 8 is a maximal irreducible subset B of 8 such that (B) is a noncyclic finite group. If B is a basic subset of 8, we call B a base of (W, 8) and (B) a basic subgroup of W. The theorems in this section are proved in our paper [8].
Theorem 5.1. (Basic Matching Theorem) Let W be a finitely generated Coxeter group with two sets of Coxeter generators 8 and 8'. Let B be a base of (W, 8). Then there is a unique irreducible subset B' of 8' such that [(B), (B)] is conjugate to [(B'), (B')] in W. Moreover,
(1) The set B' is a base of (W, 8'), and we say that Band B' match. (2) If I(B)I = I(B')I, then Band B' have the same type and there is an isomorphism
219
Theorem 5.2. Let (W, S) be a Coxeter system oj finite rank. Let B be a base oj (W, S) oj type C 2q + 1 Jor some q ~ 1, and let a, b, c be the elements oj B such that m(a, b) = 4 and m(b, c) = 3. Then W has a set oj Coxeter generators S' such that B matches a base B' oj (W, S') oj type B 2q + 1 iJ and only iJ N(a) = B U B.l. Theorem 5.3. Let B be a base oj (W, S) oj type C 2q + 1 Jor some q ~ 1, and let a, b, c be the elements oj B such that m(a, b) = 4 and m(b, c) = 3. Suppose that N(a) = B U B.l. Let d = aba, and let z be the longest element oj (B). Let S' = (S - {a}) U {d,z} and B' = (B - {a}) U {d}. Then S' is a set oj Coxeter generators Jor W such that (1) The set B' is a base oj (W, S') oj type B 2q +1 that matches B, (2) (B').l = B.l U {z}, and {z} is a component oJ (B').l, (3) N(d) = N(z) = B' U (B').l, (4) the basic subsets oj Sand S' are the same except Jor Band B'.
Theorem 5.4. Let (W, S) be a Coxeter system oj finite rank, and let B = {a, b} be a base oj (W, S) oj type D 2 ( 4q + 2) Jor some q ~ 1. Then W has a set oj Coxeter generators S' such that B matches a base B' oj (W, S') oj type D 2 (2q + 1) iJ and only iJ either v = a or v = b has the property that N(v)=BUBJ... Theorem 5.5. Let B = {a, b} be a base oj (W, S) oj type D 2 ( 4q + 2) Jor some q ~ 1. Suppose that N(a) = B U B.l. Let c = aba and let z be the longest element oj (B). Let S' = (S - {a}) U {c, z} and B' = {b, c}. Then S' is a set oj Coxeter generators oj W such that
(1) The set B' is a base oj (W, S') oj type D2(2q + 1) that matches B, (2) (B').l = BJ.. U {z}, and {z} is a component oJ (B').l, (3) N(c) = N(z) = B' U (B').l, (4) the basic subsets oj Sand S' are the same except Jor Band B'.
6. The Even Isomorphism Invariant Let (W, S) be a Coxeter system of finite rank. A pair of elements (a, b) of S is said to be unreduced if (a, b) satisfy the conditions of Theorem 5.4, that is, m( a, b) == 2 mod 4 and either v = a or v = b has the property that N(v) = {a,b} U {a,b}.l; otherwise the pair (a, b) is said to be reduced. Let (a, b) be a pair of elements of S. If (a, b) is unreduced, define m(a, b) = 2, and if (a, b) is reduced, define m(a, b) = m(a, b).
220
Let Ne be the normal closure in W of all the elements of the form ab with a and b elements of B such that m(a, b) is odd together with all the elements of the form (ab)2 such that (a, b) is an unreduced pair of elements of B. Let We = WINe, let 'r/ : W ---7 We be the quotient homomorphism, and let Be = 'r/(B). Proposition 6.1. The pair (We, Be) is a Coxeter system such that if sand t are in B, then 'r/(s) = 'r/(t) if and only if sand t are conjugate in W. If sand tare nonconjugate elements of B, then the order of'r/(s)'r/(t) is the greatest common divisor of the set of all m( u, v) such that u and v are in Band u is conjugate to s and v is conjugate to t. In particular, the order of'r/(s)'r/(t) is either even or 00. Proof. The system (We, Be) can be obtained from (W, B) by a sequence of elementary quotient operations. First we reduce to 2 all the edge labels of unreduced edges (a, b) to obtain a Coxeter system with Coxeter matrix M = (m(s, t))8,tES. Then we eliminate all the odd labelled edges. Each element of the form either st, with m(s,t) odd, or (st)2, with (s,t) unreduced, is in the commutator subgroup of W. Therefore abelianizing W factors through the quotient WINe, and so 'r/(s) = 'r/(t) if and only if sand t are in the same odd component of the P-diagram of (W, B). Hence 'r/(s) = 'r/(t) if and only if sand t are conjugate in W by Prop. 3, p. 12 of Bourbaki [3J. Suppose sand t are nonconjugate elements of Band u and v are in B, with m(u,v) < 00, and u is conjugate to s and v is conjugate to t. Then u and v are not conjugate, and so m( u, v) is even. In the coalescence of two such edges, the greatest common divisor is even. Therefore the order of 'r/(s)'r/(t) is the greatest common divisor of the set of all m(u,v) such that u and v are in Band u is conjugate to s and v is conjugate to t. D
221
Let (W, S) be a Coxeter system of finite rank. A base B of (W, S) is said to be of odd type if there are elements a and b in B, with m( a, b) odd. A base B of (W, S) is said to be reduced if for every set of Coxeter generators S' of W, the base B matches a base B' of (W,S') with I(B)I:::; I(B')I. Note that a base B of (W, S) is unreduced precisely when B satisfies the conditions of either Theorem 5.2 or 5.4. In particular, a base B = {a, b} of rank 2 is unreduced if and only if the pair (a, b) is unreduced. Theorem 6.1. Let (w, S) be a Coxeter system of finite rank. Then Ne is a characteristic subgroup of W that does not depend on the choice of Coxeter generators s. Proof. Observe that Ne is the normal closure in W of the commutator subgroups of all the basic subgroups (B) of (W, S) such that the base B is either of odd type or unreduced of rank 2. Let S' be another set of Coxeter generators of W. By the Basic Matching Theorem and Theorem 5.4, the group Ne is also the normal closure in W of the commutator subgroups of all the basic subgroups (B') of (W, S') such that the base B' is either of odd type or unreduced of rank 2. Therefore Ne is the normal closure in W of all the elements of the form s't' with s' and t' in S' and m(s', t') odd together with all the elements of the form (S't,)2 with (s', t') an unreduced pair of elements of S'. Thus Ne is a characteristic subgroup of W that does not depend on the choice of Coxeter generators S. D Let (W, S) be a Coxeter system of finite rank. P. Bahls proved in his Ph.D. thesis [1] that any finitely generated Coxeter group has at most one P-diagram, up to isomorphism, with all edge labels even. Therefore the isomorphism type of the P-diagram of (We, Se) is an isomorphism invariant of W by Theorem 6.2. We call the isomorphism type of the P-diagram of (We, Se) the even isomorphism invariant of W. For example, the even isomorphism invariant of the system D2(6) is the isomorphism type of the P-diagram of the system Al x A 1 .
222 7. Basic Characteristic Subgroups
Let (W, S) be a Coxeter system of finite rank, and let F be a family of finite irreducible Coxeter system isomorphism types. Let N(F) be the normal closure in W of the commutator subgroups of all the reduced basic subgroups of (W, S) of isomorphism type contained in F together with the commutator subgroups of all the unreduced basic subgroups of (W, S) that match a reduced basic subgroup of another system (W, S') of isomorphism type contained in F. Let W(F) = W/N(F) , let ry : W ---) W(F) be the quotient homomorphism, and let S(F) = ry(S). Theorem 7.1. The pair (W(F), S(F)) is a Coxeter system that can be obtained from (W, S) be a finite series of elementary edge quotient operations. The group N(F) is a characteristic subgroup of W that does not depend on the choice of Coxeter generators S. Proof. Quotienting out the commutator subgroup of a basic subgroup (B) of (W, S) can be realized by reducing all the even labelled edges of the Cdiagram of ((B), B) to 2 and eliminating all the odd labelled edges of the Cdiagram. Therefore (W(F), S(F)) is a Coxeter system that can be obtained from (W, S) be a finite series of elementary edge quotient operations. Let S' be another set of Coxeter generators of W. By the Basic Matching Theorem, N(F) is also the normal closure in W of the commutator subgroups of all the reduced basic subgroups of (W, S') of isomorphism type contained in F together with the commutator subgroups of all the unreduced basic subgroups of (W, S) that match a reduced basic subgroup of isomorphism type contained in F. Thus N(F) is a characteristic subgroup of W that does not depend on the choice of Coxeter generators S. 0
We call a subgroup of W of the form N(F) a basic characteristic subgroup. Corollary 7.1. If W is a finitely generated Coxeter group and N(F) is a basic characteristic subgroup of W, then the isomorphism type of W(F) = W/N(F) is an isomorphism invariant of W.
Note if F is the family of all finite irreducible Coxeter system isomorphism types, then N(F) is the commutator subgroup of W, and W(F) is the abelianization of W.
223 8. The Spherical Rank Two Invariant B. Miihlherr [7] has announced a solution of the isomorphism problem for finitely generated Coxeter groups W such that W has no basic subgroups of rank greater than 2 with respect to some set of Coxeter generators. By the Basic Matching Theorem, if W has no basic subgroups of rank greater than 2 with respect to some set of Coxeter generators, then W has no basic subgroups of rank greater than 2 with respect to every set of Coxeter generators. Therefore it makes sense to say that W has no basic subgroups of rank greater than 2 without regard to a set of Coxeter generators. In this section we describe a characteristic subgroup N2 of a finitely generated Coxeter group W such that W2 = W/N2 is a Coxeter group with no basic subgroups of rank greater than 2 and such that the isomorphism type of W 2 is an isomorphism invariant of W. Let Xn be one of the finite irreducible Coxeter systems An, Bn, Cn, E 6 , E 7, Es, F 4, G 3 , G4 of rank n 2: 3. We now define a characteristic subgroup N(Xn) of Xn for each X n . Let N(Xn) be the commutator subgroup of the Coxeter group Xn if n 2: 5 or if Xn = A 4 , G 3 , or G 4 • Let al, a2, a3 be the Coxeter generators of A3 indexed so that m(al, a2) = m(a2, a3) = 3. Let N(A3) be the normal closure in the group A3 of the element ala3' Then N(A3) is a characteristic subgroup of A3 characterized by the property that N(A3) is the unique normal subgroup of A3 such that A3/N(A3) is isomorphic to A2 according to Table 3 of Maxwell [6]. Let bl , b2 , b3 , b4 be the Coxeter generators of B4 indexed so that
Let N(B4) be the normal closure in the group B4 of the elements bl b2 and b2 b3 . Then N(B4) is a characteristic subgroup of B4 characterized by the property that N(B4) is the unique normal subgroup of B4 such that B4/N(B4) is isomorphic to A2 according to Table 3 of Maxwell [6]. Let Cl, C2, C3 be the Coxeter generators of C 3 such that m(cl, C2) = 3 and m( C2, C3) = 4. Let N (C 3 ) be the normal closure in the group C 3 of the element (C2C3)2. Then N(C 3) is a characteristic subgroup of C 3 characterized by the property that N(C 3) is the unique normal subgroup of C 3 such that C3/N(C3) is isomorphic to A2 x Al according to Table 3 of Maxwell [6]. Let Cl, C2, C3, C4 be the Coxeter generators of C 4 indexed so that
224
Let N(C 4 ) be the normal closure in the group C4 of the element CIC3. Then N(C 4 ) is a characteristic subgroup of C 4 characterized by the property that N(C4) is the unique normal subgroup of C 4 such that C 4/N(C 4 ) is isomorphic to A2 x Al according to Table 3 of Maxwell [6]. Let fl, 12,13, f4 be the Coxeter generators of F 4 indexed so that
m(fl, h) = m(h, f4) = 3 and
m(h, h) = 4.
Let N(F4) be the normal closure in the group F4 of the element (1213)2. Then N (F 4) is a characteristic subgroup of F 4 characterized by the property that N(F4) is the unique normal subgroup of F4 such that F4/N(F4) is isomorphic to A2 x A2 according to Table 3 of Maxwell [6]. Let (W, S) be a Coxeter system of finite rank. Let N(W) be the normal closure in W of the subgroups N( (B)) defined above for every base B of (W, S) of rank greater than 2. Let W(2) = W/N(W). Let ry : W --4 W(2) be the quotient homomorphism, and let S(2) = ry(S). Theorem 8.1. The pair (W(2), S(2») is a Coxeter system that can be obtained from (W, S) be a finite series of elementary edge quotient operations. The group N(W) is a characteristic subgroup of W that does not depend on the choice of Coxeter generators S.
Proof. Quotienting out the group N( (B)) for each base B of (W, S) of rank greater than 2 can be realized by elementary edge quotient operations. Therefore (W(2), S(2») is a Coxeter system that can be obtained from (W, S) be a finite series of elementary edge quotient operations. Let S' be another set of Coxeter generators of W. By the Basic Matching Theorem and the characteristic properties of the groups N(Xn), the group N(W) defined in terms of the generators S is the same as the group N(W) defined in terms of the generators S'. Thus N(F) is a characteristic subgroup of W that does not depend on the choice of generators S. D Corollary 8.1. If W is a finitely generated Coxeter group, then the isomorphism type of W(2) = W/N(W) is an isomorphism invariant of W. It may happen that (W(2), S(2») has a base of rank greater than 2. To get a quotient system with no bases of rank greater than 2, we may have to quotient out N(W(2»), and then perhaps repeat the above quotienting process several times. This leads to a finite nested sequence
225 of characteristic subgroups of W such that if WCi) = W/N(i)(W) and if 'rJi : W ----> WCi) is the quotient homomorphism, then
N Ci+1)(W)
= 'rJ;-l(N(WCi)))
for each i = 1, ... , £ - 1, and WCe) has no basic subgroups of rank greater than 2, and £ is as small as possible. We have that WCHl) = (WCi))C2) for each i = 1, ... , £ - 1. Therefore the isomorphism type of W(i) for each i = 1, ... , £ is an isomorphism invariant of W. It follows from the Basic Matching Theorem that £ does not depend on a choice of Coxeter generators for W, and so £ is an isomorphism invariant of W. We call £ the spherical rank 2 class of W. We have £ ~ 1 with £ = 1 if and only if W has no bask subgroups of rank greater than 2. Figure 2 shows the P-diagrams of a sequence WCl), ... , WCl) with £ = 4 for the Coxeter group W = WCl). Define N2 = N(£)(W). Then N2 is a characteristic subgroup of W such that W 2 = W / N2 has no basic subgroups of rank greater than 2. The isomorphism type of W 2 is an isomorphism invariant of W which we call the spherical rank 2 isomorphism invariant of W. Let 'rJ : W ----> W2 be the quotient homomorphism, and let S2 = 'rJ(S). Then (W2' S2) is a Coxeter system that can be obtained from (W, S) by a finite series of elementary edge quotient operations.
2
3 3
2
2
3
3
3
3
Figure 2
226 9. Conclusion
Let (W,8) be a Coxeter system of finite rank. In this paper, we have described three characteristic subgroups Nb, N e , N2 of W each leading to a quotient isomorphism invariant of W. It is interesting to note that N2 ~ Ne ~ Nb,
and so the quotient isomorphism invariants corresponding to Nb, N e , N2 are progressively stronger. The algorithm for finding a P-diagram for the system (Wb, 8b) starting from a P-diagram of (W, 8) is computational fast. The algorithm for finding a P-diagram for the system (We, Se) is slower since it has to determine the bases of (W, 8) of type D 2 (4q + 2) that satisfy the conditions of Theorem 5.4; but, this algorithm is only slightly slower since the conditions of Theorem 5.4 are easy to check. The algorithm for finding a P-diagram for the system (We, Se) would most likely be incorporated into an efficient computer program that determines if two finite rank Coxeter systems have isomorphic groups, since the even isomorphism invariant would usually determine that two random finite rank Coxeter systems have nonisomorphic groups. The algorithm for finding a P-diagram for the system (W2,8 2 ) is the slowest, but it is not much slower, since it only has to find a subdiagram of the P-diagram of (W, 8) of type A 3 , C 3 or G 3 before it performs an edge quotient operation on an edge of the subdiagram, and therefore reduces the complexity of the P-diagram. If the sub diagram is of type A3 or G 3 , then the edge with label 2 is eliminated. If the subdiagram is of type C 3 , then the 4 edge label is reduced to 2. The algorithm then repeats the routine of searching for a subdiagram of type A 3 , C 3 or G 3 and performing the corresponding edge quotient operation. The algorithm for finding a P-diagram for the system (W2 ,82 ) would most likely be useful in an efficient program that determines if two finite rank Coxeter systems have isomorphic groups, since the solution of the isomorphism problem for finite rank Coxeter systems that have no bases of rank greater than 2 is considerably simpler than any general solution of the isomorphism problem.
227
References l. P. Bahls, Even rigidity in Coxeter Groups, Ph.D. Thesis, Vanderbilt Univer-
sity,2002. 2. P. Bahls and M. Mihalik, Reflection independence in even Coxeter groups, Geometriae Dedicata 110 (2005), 63-80. 3. N. Bourbaki, Groupes et algebres de Lie, Chapitres 4, 5, et 6, Hermann, Paris, 1968. 4. H.S.M. Coxeter, The complete enumeration of finite groups of the form R; = (RiRj)kii = 1, J. London Math. Soc. 10 (1935), 21-25. 5. H.S.M. Coxeter, Regular Polytopes, Dover, New York, 1973. 6. G. Maxwell, The normal subgroups of finite and affine Coxeter groups, Proc. London Math. Soc. 76 (1998), 359-382. 7. B. Miihlherr, The isomorphism problem for Coxeter groups, In: The Coxeter Legacy: Reflections and Projections, Edited by C. Davis and E.W. Ellers, Amer. Math. Soc., (2006), 1-15. 8. M. Mihalik, J. Ratcliffe, and S. Tschantz, Matching theorems for systems of finitely generated Coxeter groups, Algebr. Geom. Topol. 7 (2007), 919-956.
Localization and I A-automorphisms of finitely generated, metabelian, and torsion-free nilpotent groups Marcos Zyman *
Department of Mathematics, The City University of New York-BMCC New York, New York 10007, USA E-mail: [email protected] Given a nilpotent group G and a prime p, there is a unique p-local group G(p) which is, in some sense, the "best approximation" to G among all p-local nilpotent groups. G(p) is called the p-localization of G. Let I A( G) be the group of automorphisms of G that induce the identity on G/[G, G]. IA(G) turns out to be nilpotent so its p-localization exists. Two groups are said to be in the same localization genus if their p-localizations are isomorphic for all p. We prove that if two finitely generated, torsion-free nilpotent, and metabelian groups lie in the same localization genus, their I A-groups also lie in the same localization genus. The method of proof involves basic sequences and commutator calculus.
Keywords: Nilpotent groups, p-localization, I A-automorphisms.
1. Introduction
The objective of this paper is to investigate the interaction between the I Aautomorphisms and p-localization of finitely generated, torsion-free nilpotent, and metabelian groups. In particular we prove that if two such groups lie in the same localization genus, their I A-groups also lie in the same localization genus. The proof requires an understanding of basic commutators and commutator calculus. Before stating the main theorem precisely, we discuss some initial notions and facts (see Refs. 1, 2, and 3). A group G is called p-local if the map x f--+ xn from G to itself is a bijection for n relatively prime to p. For every nilpotent group G there is a homorphism of nilpotent groups e : G -4 G(p) which p-localizes G . This means that G(p) is p-local, and for every p-local nilpotent group K, the map e* : Hom (G(p),K) -4 Hom(G,K) given by e*(
228
229 Observe that Z(p) = {min: m,n E Z and (n,p) = I} is the p-localization of the additive group of integers, and e : Z ----? Z(p) is the obvious embedding. The nilpotency class of G(p) never exceeds that of G and p-localization is a functor from the category of nilpotent groups to itself. Furthermore, the restriction of e to each term of the lower central series of G also gives a p-localization map. Finally, two nilpotent groups are said to be in the same localization genus if their p-localizations are isomorphic for every prime p. The group of I A-automorphisms of G is the subgroup of AutG consisting of those automorphisms that induce the identity on GIG', where G' is the commutator subgroup of G. We are now in a position to state the main theorem: Theorem 1.1. Let G and H be finitely generated, metabelian, and torsionfree nilpotent groups that lie in the same localization genus. Then I A( G) and I A( H) are finitely generated and torsion-free nilpotent groups, which also lie in the same localization genus.
Theorem 1.1 can be applied to the following examples, which we discuss further in §4. In the category of class 4 nilpotent groups consider Remeslennikov's groups Fs
= (x, y; [y, x, y, y]3[y, x, x, y][y, x, x, X]2),
and Fr = (x,y; [y,x,y,y]6[y,x,x,y][y,x,x,xj).
It is the case that Fs and Fr are non-isomorphic but they lie in the same localization genus. These groups satisfy the conditions of Theorem 1.1, so their I A-groups also lie in the same localization genus. Providing an additional source of examples, Pickel and Roitberg (see Ref. 4) describe a family of non-isomorphic class 4 nilpotent groups that also satisfy the hypothesis of Theorem 1.1 and prove that they lie in the same localization genus. Again, our theorem can be applied to their I A-automorphisms. We also argue in §4 that Theorem 1.1 is non-trivial by carrying out explicit computations of the inner automorphisms of G, as well as of I A( G) where G is free nilpotent of class 2 and rank 3. These calculations give that I nnG =1= I A( G) in general.
I am indebted to Joseph Roitberg, my advisor, who suggested the problem and guided me through the research process; as well as to Gilbert
230 Baumslag for all his help and support. I also wish to thank Marianna Bonanome, Margaret H. Dean, and Katalin Bencsath for many helpful conversations. During the early stages of my research I used the group theory package MAGNUS (available at http://sourceforge.net/projects/magnus), developed by the Center for Algorithms and Scientific Software at the City College of New York. This paper is dedicated to Joseph Roitberg. 2. Preliminary discussion
For any group G let "fiG denote the i-th term of the lower central series, where "f1 = G and "f2G is the commutator subgroup of G. The following result of P. Hall (see Ref. 5) is the departing point of our discussion. Theorem 2.1. Let G be nilpotent of class c. Then • "fjIA(G) transforms each "fiGhi+jG identically, and • I A( G) is nilpotent of class c - 1.
The fact that G modulo its center is isomorphic to the group of inner automorphisms of G also suggests that IA(G) may be torsion-free if Gis torsion-free. In fact, the following is true: Lemma 2.1. If G is a finitely generated, torsion-free nilpotent group, I A( G) is also finitely generated, and torsion-free nilpotent.
Lemma 2.1 can be proved by induction on the class of G (see Lemmas (1.2.9) and (1.2.10) in Ref. 6). Assuming nilpotency of G is crucial here. C.K. Gupta proved that I A( G) is metabelian if G is a two generator metabelian group (see [7J in Ref. 7). However, IA(G) need not be finitely generated (see Theorem C in Ref. 7). As a corollary of Theorem 2.1 and Lemma 2.1, we have: Corollary 2.1. If G is finitely generated and torsion-free nilpotent of class c, then I A( G) is finitely generated and torsion-free nilpotent of class c - 1.
The following definitions and remarks (see Ref. 2) lead to a very useful reformulation of Theorem 1.1. Let
231
there is an n, relative prime to p, such that hn belongs to the image of cpo A homomorphism satisfying (ii) is called p-surjective . The proof of Theorem 1.1 depends on the fact that a homomorphism of nilpotent groups cp : G ---; H is a p-localization map if, and only if: (1) His p-local, and (2) cp is a p-isomorphism.
In fact, we actually prove the following statement, from which Theorem 1.1 readily follows: Theorem 2.2. Let G be finitely generated, metabelian, and torsion-free nilpotent. Then I A(G(p)) is p-local, and the natural map
is a p-isomorphism. The motivation for Theorem 2.2 comes from a related result by Maruyama (see Ref. 8) in homotopy theory. To describe it, let X be a simply connected CW-complex and denote by co(X) the group of homotopy classes of self-homotopy equivalences of X that induce the identity on all homology groups. E. Dror and A. Zabrodsky proved that co(X) is nilpotent (see Ref. 9), so its p-localization makes sense. Maruyama's result is that the homomorphism co(X) ---; co(X(p)) obtained by localizing each homotopy class is in fact the localization homomorphism of nilpotent groups co(X) ---; co(X)(p). Theorem 2.2 does not follow from Maruyama's result since X is assumed to be simply connected, and the elements of co(X) induce the identity an all homology groups. 3. Proof of the main theorem
3.1. Powers of IA-automorphisms We first develop a technique to compute powers of I A-automorphisms. Let G be the p-localization of a finitely generated, torsion-free, metabelian, and nilpotent group of class c. Let X = {Xl, ... ,xr } be a finite set that generates G as a p-local group. Then G is also generated, as a p-local group, by the set of basic commutators B = {b l , ... , bm } on X; where bi = Xi for i = 1, ... , r. Since G is metabelian, any basic commutator on B is of the form
232
Denote the weight of bi by wt(b i ). Let
where wt(b i ) ::::; C - 1 and Ai E G'. Ai can be expressed as a product of rational powers of basic commutators of weight at least 2, and at most c. So we can write Ai
=
II [bk,Xz]* k>l
where wt(bk) ::::; C - 1 for each k; and v(i) is relatively prime to p. Direct computations involving commutator calculus give
where Dil =
II([bk,Ad[Ak,xz])*&. k>l
In general, for each i, we may construct a sequence of elements of G':
(1) where Dil
= II([b k , Az][Ak,xz])*, k>l
Dij
= II([h,Dl(j-l)][Dk(j-I),xd)~
for j > 1,
k>l
(()m(A.) _ A.tUil ,Cl(m) ,C2(m) ..t Ui2
where Cj(m)
=
rc",(m)
•.. Uim
(m), = "(mm!_ ')" J
J.
J,
233
Further computation and Lemma 3.1 give: Lemma 3.2. = A.rp(k)rp2(k) ... rpm-l(k) = b-Am8dl(m)8d2(m) ...•dm - 1 (m) rp m(b) t t t t t t t tl t2 U i( m-l) ,
where
3.2. Proof that I A( G) is p-local if G is p-local Consider the map
IA(G) rp
~ f--+
IA(G) rpn
where (n,p) = 1, and G is the p-localization of a finitely generated, metabelian, and torsion-free nilpotent group of class c. We now embark on the proof that this map is a bijection. Lemma 3.3.
IA(G)
~
IA(G)
is one-to-one.
Proof. To see this, let
rpn = '1f;n. We wish to prove that rp
= '1f;.
For this purpose put
rp(b i ) = biAi,
where 1 :::; wt(b i ) :::; C - l. As usual we write
(2) and
(3)
234
In order to show that cp = 'Ij; we proceed by reverse induction on wt(b i ). Suppose wt(b i ) = c - 1. Then both Ai and ..t belong to IC' cpn(b i ) = 'lj;n(bi ) implies that biAi = buii, so that Ai = Ai. By p-locality this means that Ai = Ai. Now suppose that bi satisfies
Assume the induction hypothesis that cp = 'Ij; on 1}+1. Our goal is to show that cp(b i ) = 'Ij;(b i ). Since cpn(b i ) = 'lj;n(b i ) then b.An8d,(n) ... 8dn-,(n) = b.AnJdl(n) ... Jdn-l(n) , , >1 i(n-l) ',>1 i(n-l) . Using the fact that we are in a p-local group, we obtain: A_I -1 A dl(n) -1 A dn_l(n) AiAi = (8i1 8il ) n ... (8 i (n_l)8 i(n-l)) n .
(4)
Since wt(b i ) = j, Ai and Ai each belongs to Ij+!' In fact, by Eq. 4, the product AiAi1 actually lies in 1}+2. By induction:
cp(Ai)
= 'Ij;(Ai) = A i Ji1 .
This means that A_I
cp(AiAi )
A A
= Ai8il(Ai8il)
-1
A_I
A_I
= AiAi 8i1 8i1
.
Simply because cp is and I A-automorphism, it follows that A_I
8i1 8i1
E 1}+3.
Similarly, if we evaluate cp on any 8imJ~ in the right hand side of Eq. 4 we have (again by induction on j) that
(5) since
Again, because cp is and I A-automorphism, Eq.
5 implies that
8i (m+!) Ji(;;'+I) is in a higher commutator subgroup than the commutator subgroup where 8imJ~ lies. Hence the entire right-hand side of Eq. 4 belongs to
1}+3.
We have established that A_I
AiAi
E 1}+3.
Apply the same argument repeatedly to finally conclude that A_I
AiAi
E Ie+!
= 1.
235 This completes the proof that I A( G)
-+
I A( G) is one-to-one.
o
Lemma 3.4. Let G be the p-localization of a finitely generated, metabelian, and torsion-free nilpotent group of class c, and let n be relatively prime to p. Then the map from I A( G) to itself given by cp I - ' cpn is onto. Proof. Let {b 1 , b2 , ... } be the basic commutators on X = {Xl, •.. , x r } of weight at most c - 1. Let cp(b i ) = biAi E I A( G). Let Oil, Oi2, ... ,Oi(c-2) be the o-sequence associated to Ai' (Since G has class c, we can henceforth assume that Oil = 1 for I > c - 2). We claim that there are p-local integers E1 (n), ... , Ec- 2 (n) that depend on n (and c), such that
• ·'·(b) = b·A;\:o
Ai
=
II[bk, Xzl ikl , k>l
where the ikl are p-local integers. We wish to find p-local integers such that
E1, ... ,E c -2,
depending on nand c alone,
1
gives an I A-automorphism where Qi = A; oa ... 0:(;~2) and 1/Jn = cpo We will show that these E'S can in fact be found by "solving" the equation
1/Jn
cpo Associated with =
we have the corresponding sequence of deltas:
Qi
Ji1 , ... , Ji (c-2)' Commutator calculus gives:
o"(b·) 'f' 2
1
= b -A n oEt ........... ·0
.I'(A-) 'f' 2
2
2
1
= A-ond 0<1 ......... 0«C-3 c-2 ) 2
22
2
1
1/J(Oi1) = Oi10i~0:j ...... O:(;~2) 1
1/J( Oi2)
Additional calculations yield:
=
Oi20Z~ 0:4... 0:(;~2)
236 1/ 15il
n2
15"'1 ... 15"'c-3 and a J' depends on i2 i(c-2)'
n,EI,···,Ej.
• 'IjJ(Jid
=
i3c - 4 151/n315i31 .. . 15i(c-2)' and f3J' depends on i2 i3
J iI J i2 where J i2
We continue to find similar expressions for J ij in terms of the 15ij , the last expression being (l/n)c-1
A
15i (c-2) We now show that the
E'S
=
15i (c-2)
appearing in the equation
(6) can be chosen so that they depend only on n. For this purpose we use the formula that describes powers of IA-automorphisms (see Lemma 3.2). Equation 6 becomes: A15n€1 ... 15 nfc - 2 Jd 1 ... Jd(c-2) = A t tl
t(c-2) tl
t(c-2)
t·
Canceling Ai and rewriting the J's in terms of the 15's yields: 8nE1 , .. 8 n €c-2 i1 i(c-2)
2 2-1 1 i2 , .. 8"'c-3 )d 1(8-;;J8f31 ... 8f3c-4 )d , .. (8 nc - 1 )d i1 i(c-2) i2 i3 i(c-2) i(c-2) -, (8~8"'1 1
Now solve for each
Ei
1
C
_
in the following way:
nEI + ,&dl = 0 =} EI = - ~dl' which means that EI can be chosen so that it depends on n only. • nE2 + aId l + Ad 2 = 0 =} E2 = - lna I dl - Ad 2. Since al depends on n n n and EI, we conclude that E2 can be chosen so that it depends on n only. • nE3 + a2dI + f3 l d 2 + Ad 3 . Again, 3 = 0 =} E3 = -la n n 2 d l - lf3 n l d 2 - Ad n a2 depends on n, EI, and E2; so E3 can be chosen in such a way that it depends on n alone. •
Continue this process to choose EI, ... , Ec -2 so that they depend only on n. Computations show that 'IjJ lifts to a well-defined I A-automorphism (see the second part of the proof of theorem (2.2.2) in Ref. 6). Lemma 3.4 is ~~~.
3.3. IA(G)
0
-+
IA(G(p»
is a p-isomorphism
Let G = gp(XI, . .. , xm) be a finitely generated, metabelian, and torsionfree nilpotent group of class c. Let G(p) be its localization at the prime p.
237 Consider the localization diagram
G ~ G fp -----7
G(p)
and observe that if! E IA(G), then !p E IA(G(p»). Lemma 3.5. The homomorphism
is a monomorphism. Proof. Let! E ker (IA(G) -+ IA(G(p»)) and put !(Xi) = XiAi where Ai lies in G' = [G, G]. For 9 E G, write e(g) = g. If Xi is a generator of G, Xi belongs to G(p) and
Xi = !P(Xi) = !(Xi) = xiAi. Hence, Ai
=
I in G~p), which means that Ai belongs to ker
(e :G'
-+
G~P»)'
Since G' is torsion-free and e : G' -+ G~p) is a localization map (see §1), e is one-to-one so that Ai = 1. Hence !(Xi) = Xi; and therefore
is indeed a monomorphism.
o
Remark. The discussion that led to the b-sequence (see 1 in § 3) and Lemma 3.2 remains valid for I A( G), where G is finitely generated, torsionfree nilpotent, and met abelian (the condition of G being p-local is dropped). This observation will be used in the sequel. Lemma 3.6. IA(G)
-+
IA(G(p») is p-surjective.
Proof. Let
238 there exists an integer Si, relatively prime to p, such that the image of e : G' -. G~p) for each i = 1,2, ... , m. Put
A:i belongs to
A~l clearly lies in the image of e : G' -. G~p) (for each i = 1,2, ... , m)
because such image is a subgroup of G~p)' Similarly, choose (}2,···, ()c-l (independent if i) so that O~:+l belongs to the image of e : rk+2(G) -. rk+2(G(p)), for k = 1, ... , c - 2. Let
We have the following: (1) () is relatively prime to p, (2) Ai belongs to the image of e : G' -. G~p), and (3) Oik belongs to the image ofe : rk+2(G) -. rk+2(G(p)), (k
= 1, ... ,c-2).
Using Lemma 3.2, we see that •
•
- (a) _ d1 ( () ) -2_ d2 (() ) -- (a) 3 _ d3 (() ) -- (a) 4 -
. •
•
a(a-l) 2 a(a-1)(a-2) 3! a(a-1)(a-2)(a-3) 4!
.
- ( a ) dc-2 () () c-1
_ -
a(a-1)(a-2) ... (a-c+2) (c-1)! .
Fix 1 :::; j :::; c - 2 and consider the number
. _ ( () ) _ ()(()-1)(()-2) .. ·(()-j) dJ (()) j +1 (j + I)! . Write
where p and Ej are relatively prime. (If p does not divide (j (Xj = 0.) Next, let
Notice that p and
E
are relatively prime. Now put S = E().
+ I)!
take
239 Again, p and s are relatively prime. Invoking Lemma 3.2 once more, we see that for each j:
dj(s) =
f1 f 2··· f c_2 U (s-1)(s-2)···(s-j)
=
pOi Jfj f1f2··· fj-1fJ+1 ... fC-2U(S -
l)(s - 2)··· (s
-
j)
pOij
As dj(s) is an integer, and pOij does not divide f1f2··· fj-1fJ+1··· fc-2U, has to divide (s - l)(s - 2)··· (s - j). Hence, each integer s,d 1 (s),d 2 (s), ... ,d c - 2 (s) is a multiple of u. The crucial conclusion is that
pOij
(1) s is relatively prime to p, (2) Ai belongs to the image of e : G'
-->
G~p), and
(3) 151;(8) lies in the image of e : I'J+2(G) c - 2.
-->
I'J+2(G(p)), for j
= 1,2, ... ,
We can therefore choose C¥i E G' such that ai = Ai, and Dij E I'J+2 (G) 8 such that Dij = 151/ ) for each j = 1,2, ... , c- 2; where Ii = e(g) for 9 E G. Using Lemma 3.2 yet again we see that
cp
8(-.) _ - -A8 r5d1 (8) ... r5dC-2 (8) X. - x. • i1 i(c-2) .
Let fJi
= C¥i D i1 ... D i (c-2)
E
G'.
Define the following map on the generators of G: f(Xi)
=
xifJi·
It is routine to show f can be extended to an element of I A( G) (see p. 47 of Ref. 6). Then, fp and cp8 coincide on the p-generators of G(p) and will 0 therefore be equal as I A-automorphisms. The proof of Theorem 2.2 is now complete, and Theorem 1.1 readily follows. 4. Examples
4.1. An example where InnG
=I IA(G)
Let (denote the center of G. If it were the case that InnG = IA(G) for all nilpotent groups G, Theorem 1.1 would be trivial since we would have
240
due to the facts that (G/()(p) ~ G(p)/(p) (see Ref. 2) and H/((H) ~ InnH for any group H. To demonstrate that Theorem 1.1 is nontrivial in general we compute I nnG and I A( G) where G is free nilpotent of class 2 and rank 3. It will then be clear that InnG -=f. IA(G). Let
G=(x,y,z) be free nilpotent of class 2 on the generators {x,y,z}. Put C12 = [x,y], C13 = [x,z], and C23 = [y,z] (we will use similar notations in §4.2). Then
is a basic sequence of basic commutators on {x, y, z}. Since G is free nilpotent, every g E G can be uniquely written as
I
,
I
e'
e'
e'
If g' = xelye2ze3c1~2c133c233 is another element of G, standard commutator calculus gives ,_
gg -
Xel+e~ Ye2+e; Ze3+e; Ce12+e~2-e;e2 Ce13+ 12 13
e ;3- e ;ea e23+e~3-e~e3
c23
.
(7)
Consider the following nine elements of I A( G):
az
b
z z
It is straightforward to show that
In ax ",ay Inaz Inbx ,n by ,nbz InCX In cy InCZ
..- -..-1 ..-2 ..-3 ..-4 ..-5 ..-6 ..-7 ..-8 ..-g .
241
Since I A( G) is a torsion-free abelian group, this proves that
IA(G) is free abelian of rank 9. Now choose
= g-1 xg ,
By the normal form (see Eq. 7) we find that
and further use of Eq. 7 ultimately gives
= xc~~cg,
We obtain three specific elements of InnG by setting, in turn, e l l , e2 = e3 = 0; e2 = 1, e1 = e3 = 0; and e3 = 1, el = e2 = O. These are:
= XC12,
It is straightforward that
and that this expression is unique. This proves that
I nnG is free abelian of rank 3 and hence
InnG
-# IA(G)
for G free nilpotent of class 2 and rank 3. 4.2. Remeslennikov's groups
In the category of nilpotent groups of class 4, consider Remeslennikov's groups:
and
242
It is clear that Fs and Fr are also metabelian and torsion-free. Using symmetric bilinear forms (see §3.1 in Ref. 6), it is possible to show that Fs and Fr are not isomorphic but lie in the same localization genus. Corollary 2.1 gives that IA(Fs) and IA(Fr) are torsion-free nilpotent of class 3, and hence metabelian. It now follows from Theorem 2.2 that IA(Fs) and IA(Fr ) lie in the same localization genus.
4.3. Pickel-Roitberg groups Pickel and Roitberg (see Ref. 4) associate a finitely generated nilpotent group of class 4 to a symmetric bilinear form as follows: let a : zn X zn - t Z be a symmetric bilinear form satisfying 0'( q, q) E 2Z for all q E zn. Let {e1' ... ,en} be any ordered basis of zn, and put
= 2bii , • O'(ei' ej) = O'(ej' ei) = • O'(ei' ei)
bij for i =F j.
The n x n symmetric integral matrix B associated to this form has entries 2bii in the diagonal and bij = bji , when i =F j. Two such forms are said to be in the same localization genus if they are equivalent up to a unit over the ring Z(p), for every prime p. Let F = (x, y, Z1, ... , zn) be the free metabelian and nilpotent group of class 4, on n + 2 generators. Consider the group G(O') =
(x, y, Z1, ... , Zn; p) ,
where p=
IT [y, x, Zi, Zj]b
ij •
i~j
It is not hard to show that G is torsion-free if and only if the bij are relatively prime. Pickel and Roitberg proved that if 0'1 and 0'2 are in the same localization genus, then G 1 = G(O'1) and G 2 = G(O'2) are in the same localization genus. Assuming that the forms 0'1 and 0'2 give rise to torsion-free groups, it follows that G 1 and G 2 are torsion free, nilpotent, and met abelian of class 4. Theorem 1.1 now gives that IA(G 1 ) and IA(G 2 ) are in the same localization genus.
243
References 1. P. Hilton, Localization and cohomology of nilpotent groups, Math. Z. 132
(1973), 263-286. 2. P. Hilton, G. Mislin, and J. Roitberg, Localization of nilpotent groups and spaces, Notas de Matematica, North-Holland Mathematics Studies 15 (1975). 3. R. B. Warfield, Nilpotent groups, Lecture Notes in Mathematics 513, SpringerVerlag, (1976). 4. P. F. Pickel and J. Roitberg, Automorphism groups of nilpotent groups and spaces, J. Pure Appl. Algebra 150 (2000), no. 3, 307-319. 5. P. Hall, The Edmonton notes on nilpotent groups, Queen Mary College Mathematics Notes, (1969). 6. M. Zyman, I A-automorphisms and localization of nilpotent groups, Ph.D. Thesis-City University of New York, (2007). 7. S. Bachmuth, G. Baumslag, J. Dyer, and H. Y. Mochizuki, Automorphism groups of two generator metabelian groups, J. London Math. Soc. (2) 36 (1987), no. 3, 393-406. 8. K. Maruyama, Localization of self-homotopy equivalences inducing the identity on homology, Math. Proc. Cambridge Philos. Soc. 108 (1990), nO. 2, 291-297. 9. E. Dror and A. Zabrodsky, Unipotency and nilpotency in homotopy equivalences, Topology 18 (1979), no. 3,187-197.
