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0, l' E Z, b' E B"}. Moreover, the relation a"-lb,ma" = b'n holds in L. To see this, observe that (the subscripted version) all-lb~mall = b~n holds in Land b' has the form b' = b~~1 ... b~:l, where ti = ±l. Since the b~'s commute, a"-lb,ma" = all-l(b~~1m ... b~~lm)all = 1 m II l-lb'E2ma" a"-1b'E1m II = b'E1nb'E2n b~Eln = (b'E1 b'El)n = a1I- b'E1 21 a a 22 ... 21 a 21 22 ... 2£ 21 ... 21 '"-.".-''"-.".-'
'"-.".-'
b'n. Similarly, analyzing actions inherited in analogous quotients of
H
=
T)(J
(x),
we see that T IT(l) = ( ... , Yi, ... (i E Z); ... , Yi' = Yi~l' ... (i E Z), ... , [Yi, Yj] 1, ... (i,j E Z)), whence
which implies that HIT(l) /QIT(l) = TIT(l) /QIT(l)
)(J
((xT(l»)QIT(l»)
=
78
and so
H/Q Letting M = follows, where 1, ... (i,j E Z)) {x"ky'l k E Z,
~
T/Q
) 0 'lj;) = V(¢» U V('lj;), where 0 E {V, i\, ----;}, and V(Vx¢» = V(3x¢» = V(¢» ,,{x}. We write ¢>(XI, ... ,Xn ) in the case when V(¢» (XI, ... ,Xn ) E .c(X) and ml, ... ,mn E M then one can define, following the conditions F1)-F3), the relation "¢> is true in M under the interpretation Xl ----; ml,.'" Xn ----; m n" (symbolically M F ¢>(ml, ... , m n )). It is convenient sometimes to view this relation as an n-ary predicate ¢>M on M. If h : X ----; M is an interpretation of variables then we denote ¢>h = ¢>M (h(XI)"'" h(xn)). A set of formulas h for every ¢> E . In this case one says that is realized in M. The following result is due to Malcev, it plays a crucial role in model theory. Theorem [Compactness Theorem] Let K be a class of C-structures and , 'lj; E .c(X) are called equivalent if ¢>h = 'lj;h for any interpretation h: X ----; M and any C-structure M. One of the principle results in mathematical logic states that any formula ¢> E .c( X) is equivalent to a formula 'lj; in the following form: (1)
where Qi E {V,3} and 'lj;ij is an atomic formula or its negation. One of the standard ways to characterize complexity of formulas is according to their quantifier prefix QIXI ... Qmxm in (1). If in (1) all the quantifiers Qi are universal then the formula 'lj; is called universal or V-formula, and if all of them are existential then 'lj; is existential or 3-formula. In this fashion 'lj; is V3-formula if the prefix has only one
85
alteration of quantifiers (from V to 3). Similarly, one can define 3V-formulas. Observe, that V- and 3-formulas are dual relative to negation, i.e., the negation of V-formula is equivalent to an 3-formula, and the negation of 3formula is equivalent to an V-formula. A similar result holds for V3- and 3Vformulas. One may consider formulas with more alterations of quantifiers, but we have no use of them in this paper. A formula in the form (1) is positive if it does not contain negations (i.e., all'lfJij are atomic). A formula is quantifier-free if it does not contain quantifiers. We denote the set of all quantifier-free formulas from .c(X) by qf,.c(X), and the set of all atomic formulas by At.c(X). Recall that a theory in the language £ is an arbitrary consistent set of sentences in 12. A theory T is complete if for every sentence ¢ either ¢ or -,¢ lies in T. By Mod(T) we denote the (non-empty) class of all 12-structures M which satisfy all the sentences from T. Structures from Mod(T) are termed models of T and T is a set of axioms for the class Mod(T). Conversely, if K is a class of 12-structures then the set Th(K) of sentences, which are true in all structures from K, is called the elementary theory of K. Similarly, the set Thv(K) (Th3(K)) of all V-sentences (3-sentences) from Th(K) is called the universal (existential) theory of K. The following notions play an important part in this paper. Two 12-structures M and N are elementarily equivalent if Th(M) = Th(N) , and they are universally (existentially) equivalent if Thv(M) = Thv(N) (Th3(M) = Th3(N)). In this event we write, correspondingly, M = N, M =v N or M =3 N. Notice, that due to the duality mentioned above M =v N {=} M =3 N for arbitrary 12-structures M and N. A class of 12-structures K is axiomatizable if K = Mod(T) for some theory Tin 12. In particular, K is V- (3-, or V3-) axiomatizable if the theory T is V- (3-, or V3-) theory.
3. Algebras
There are several types of classes of 12-structures that playa part in general algebraic geometry: prevariaeties, quasivarieties, universal closures, and Aalgebras. We refer to [34] for a detailed discussion on this and related matters. Here we present only a few properties and characterizations of these classes, that will be used in the sequel. Most of them are known and can be found in the classical books on universal algebra, for example, in [30]. On the algebraic theory of quasivarieties, the main subject of this section, we refer to [15].
86 3.1. Congruences
In this section we remind some notions and introduce notation on presentation of algebras via generators and relations. Let M be an arbitrary fixed C-structure. An equivalence relation () on M is a congruence on M if for every operation P E F and any elements m1, ... ,mnF , m~, ... ,m~F EM such that mi rvB m~, i = 1, ... ,np, one has pM(m1, .. " m nF ) rvB pM(m~, ... , m~F)' For a congruence () the operations pM,
P E F, naturally induce welldefined operations on the factor-set M/(). Namely, if we denote by m/() the equivalence class of mE M then pMjB is defined by pMjB(ml/(), ... , m nF /())
= pM(m1, ... , m nF )/()
for any m1,"" m nF E M. Similarly, cM / B is defined for c E C as the class cM /(). This turns the factor-set M/() into an C-structure. It follows immediately from the construction that the map h : M ---4 M / (), such that h(m) = m/(), is an C-epimorphism h : M ---4 M/(), called the canonical epimorphism. The set Con(M) of all congruences on M forms a lattice relative to the inclusion ()1 ~ ()2, i.e., every two congruences in Con(M) have the least upper and the greatest lower bounds in the ordered set (Con(M), ~). To see this, observe first that the intersection of an arbitrary set e = {()i, i E I} of congruences on M is again a congruence on M, hence the greatest lower bound for e. Now, the intersection of the non-empty set {() E Con(M) I ()i ~ () 'V ()i E e} is the least upper bound for e. The following result is easy. Lemma 3.1. Let M be an C-algebra, {()i l i E I} ~ Con(M) and iEI Bi · Then M/B embeds into the direct product DiE I M/()i via the diagonal monomorphism m/B ---4 DiE I m/()i.
() =
n
A homomorphism h : M ---4 N of two C-structures determines the kernel congruence ker h on M, which is defined by m1
rvkerh
m2
{==}
h(m1) = h(m2),
m1, m2 EM.
Observe, that if () E Con(M) and B ~ ker h then the map h : M/() ---4 N defined by h(m/()) = h(m) for m E M is a homomorphism of C-structures. Definition 3.1. A set of atomic formulas t:. ~ Atc(X) is called congruent if the binary relation ()t::. on the set of terms Tc(X) defined by (where t1,t2 E T.c(X))
87 is a congruence on the free .c-algebra T.c(X). The following lemma characterizes congruent sets of formulas. Lemma 3.2. A set of atomic formulas ~ ~ At.c(X) is congruent if and
only if it satisfies the following conditions: (1) (t = t) E ~ for any term t E T.c(X); (2) if (tl = t2) E ~ then (t2 = tI) E ~ for any terms h, t2 E Tc(X); (3) if (h = t2) E ~ and (t2 = t3) E ~ then (h = t3) E ~ for any terms tl, t2, t3 E T.c(X); (4) if (h = SI), ... ,(tnF = snF) E ~ then (F(tl, ... ,tnF ) F(SI, ... ,SnF)) E ~ for any terms ti,Si E T.c(X), i = 1, ... ,nF, and any functional symbol F E .c. Proof. Straightforward.
o
Since the intersection of an arbitrary set of congruent sets of atomic formulas is again congruent, it follows that for a set ~ ~ Atc(X) there is the least congruent subset [~] ~ Atc(X), containing ~. Therefore, ~ uniquely determines the congruence Bt::. = B[t::.]. For an .c-algebra M generated by a set M' ~ M put X = {xm I m E M'} and consider a set ~M' of all atomic formulas (tl = t2) E At.c(X) such that M F (h = t2) under the interpretation Xm -+ m, m EM'. Obviously, ~M' is a congruent set in At.c(X) (the set of all relation in M relative to M'). A subset 5 ~ ~M' is called a set of defining relations of M relative to M' if [5] = ~M" In this event the pair (X I 5) termed a presentation of M by generators X and relations 5. Lemma 3.3. If (X I 5) is a presentation of M then M ~ Tc(X)/Bs. Proof. The map h' : X -+ M' defined by h'(x m ) = m, m EM', extends to a homomorphism h : T.c(X) -+ M. Clearly, tl "'kerh t2 if and only if (tl = t 2 ) E [5] for terms t l , t2 E T.c(X). Therefore, T.c(X)/Bs ~ T.c(X)/ ker h. Now the result follows from the isomorphism T.c(X)/ ker h ~ M. 0 3.2. Quasivarieties
In this section we discuss quasivarieties and related objects. The main focus is on how to generate the least quasi variety containing a given class of structures K. A model example here is the celebrated Birkhoff's theorem which describes Var(K), the smallest variety containing K, as the class
88
HSP(K) obtained from K by taking direct products (the operator P), then substructures (the operator S), and then homomorphic images (the operator H). Along the way we introduce some other relevant operators. On the algebraic theory of quasivarieties we refer to [15] and [30]. We fix, as before, a functional language £ and a class of £-algebras K. We always assume that K is an abstract class, i.e., with any algebra M E K the class K contains all isomorphic copies of M. Recall that an identity in £ is a formula of the type
where t, s are terms in £. Meanwhile, a quasi-identity is a formula of the type
where t(x), sex), ti(X), Si(X) are terms in £ in variables x = (Xl, ... , xn). A class of £-structures is called a quasivariety (variety) if it can be axiomatized by a set of quasi-identities (identities). Given a class of £structures K one can define the quasivariety Qvar(K), generated by K, as the quasivariety axiomatized by the set Thqj(K) of all quasi-identities which are true in all structures from K, i.e., Qvar(K) = Mod(Thqj(K)). Notice, that Qvar(K) is the least quasivariety containing K. Similarly, one defines the variety Var(K) generated by K. Observe, that an identity Vx(t(x) = sex)) is equivalent to a quasiidentity V x( X = X --t t( x) = s( x)), therefore, Qvar(K) ~ Var(K). Before we proceed with quasivarieties, we introduce one more class of structures. Namely, K termed a prevariety if K = SP(K). By Pvar(K) we denote the least prevariety, containing K. The prevariety Pvar(K) grasps the residual properties of the structures from K. An £-structure M is separated by K if for any pair of non-equal elements ml, m2 E M there is a structure N E K and a homomorphism h : M --t N such that h( ml) "I=h(m2)' By Res(K) we denote the class of £-structures separated by K. In the following lemma we collect some known facts on prevarieties.
Lemma 3.4. For any class of £-structures K the following holds: 1) Pvar(K) = SP(K) ~ Qvar(K); 2) Pvar(K) = Res(K); 3) Pvar(K) is axiomatizable if and only if Pvar(K)
= Qvar(K).
89 Proof. Equality 1) follows directly from definitions. 2) was proven for groups in [34], here we give a general argument. It is easy to see that Res(K) is a prevariety, so Pvar(K) ~ Res(K). To show converse, take a structure M E Res(K) and consider the set I of all pairs (ml,m2), ml,m2 E M, such that ml =I- m2· Then for every i E I there exists a structure M E K and a homomorphism hi : M ----) M with hi(ml) =I- h i (m2). The homomorphisms hi, i E I, give rise to the "diagonal" homomorphism h : M ----) Ilo M, which is injective by construction. Hence M E SP(K), as required. 0 3) is due to Malcev [31]. Prevarieties play an important role in combinatorial algebra, they can be characterized as classes of structures admitting presentations by generators and relator. Namely, let X be a set and ~ a set of atomic formulas from c(X). Following Malcev [30], we say that a presentation (X I ~) defines a structure M in a class K if there is a map h : X ----) M such that D1) h(X) generates M and all the formulas from ~ are realized in M under the interpretation h; D2) for any structure N E K and any map f : X ----) N if all the formulas from ~ are realized in N under f then there exists a unique homomorphism g : M ----) N such that g(h(x)) = f(x) for every x E X. If (X I ~) defines a structure in K then this structure is unique up to isomorphism, we denote it by FK(X, ~). Theorem [30] A class K, containing the trivial system £, is a prevariety if and only if any presentation (X I ~) defines a structure in K. To present similar characterizations for quasivarieties we need to introduce the following operators. As was mentioned above, P(K) is the class of direct products of structures from K. Recall, that the direct product of C-structures M i , i E I, is an C-structure M = DiE! Mi with the universe M = DiE! Mi where the functions and constants from C are interpreted coordinate-wise. If all the structures Mi are isomorphic to some structure N then we refer to DiE! Mi as to a direct power of N and denote it by N!, By Pw(K) we denote the class of all finite direct products of structures from K. Recall, that a substructure N of a direct product DiE! Mi is a subdirect product of the structures M i , i E I, if pj(N) = M j for the canonical
90
projections Pj : I1iEI Mi -- Mj , j E I. By P,.(K) we denote the class of all subdirect products of structures from K. Let I be a set, D a filter over I (i.e., a collection D of subsets of I closed under finite intersections and such that if a E D then bED for any b -> and epimorphic direct limits of structures from K. The following result gives a characterization of quasivarieties in terms of direct limits. Lemma 3.6. For any class of .c-structures K the following holds:
Proof. See [15] ([Corollary 2.3.4]).
o
3.3. Universal closures In this section we study the universal closure Ucl(K) = Mod(Thv(K)) of a given class of .c-structures K. Structures from Ucl(K) are determined by local properties of structures from K. To explain precisely we need to introduce two more operators. Recall [5,34]' that a structure M is discriminated by K if for any finite W of elements from M there is a structure N E K and a homomorphism set
92
h : M ---., N whose restriction onto W is injective. Let Dis(K) be the class of .c-structures discriminated by K. Clearly, Dis(K) ~ Res(K). To introduce the second operator we need to describe local submodels of a structure M. First, we replace the language .c by a new relational language .cre!, where every operational and constant symbols F E :F and c E C are replaced, correspondingly, by a new predicate symbol RF of arity nF + 1 and a new unary predicate symbol Re. Secondly, the structure M l l . MTel MTel turns into a .c re -structure Mre , where the predIcates Re and RF are defined by R1) for m
E
M the predicate R.;tTel (m) is true in
Mrel
if and only if
eM =m;
R2) for mo, ml,"" m nF E M the predicate R-j;:lTel (mo, ml,"" m nF ) is true in Mrel if and only if FM(ml,'" ,mnF ) = mo.
Third, if .co is a finite reduct (sublanguage) of .c then by Meo we denote the reduct of Mrel, where only predicates corresponding to constants and operations from .co are survived, so Meo is an .cae/-structure. Now, following [30], by a local submodel of M we understand a finite substructure of Me o for some finite reduct .co of .c. Finally, a structure M is locally embeddable into K if every local submodel of M is isomorphic to some local submodel of a structure from K (in the language .coel ). By L(K) we denote the class of .c-structures locally embeddable into K. It is convenient for us to rephrase the notion of a local submodel in terms of formulas. Let .c' be a finite reduct of.c and X a finite set of variables. A quantifierfree formula cp in .c' is called a diagram-formula if cp is a conjunction of atomic formulas or their negations that satisfies the following conditions: 1) every formula -,(x = y), for each pair (x,y) E X 2 with x in cp;
i=
y, occurs
2) for each functional symbol F E .c' and each tuple of variables (XO,Xl, ... ,X nF ) E xnF+I either formula F(XI,".,X nF ) = Xo or its negation occurs in cp; 3) for each constant symbol c E .c' and each x E X either x = c or its negation -,(x = c) occurs in cpo We say that cp is a diagram-formula in .c if it is a diagram-formula for some finite reduct .c' of.c and a finite set X. The name of diagram-formulas comes from the diagrams of algebraic structures (see Section 3.4).
93
The following lemma is easy. Lemma 3.7. For any local submodel N of M there is a diagram-formula 'PN(X) in a finite set of variables X of cardinality [N[ such that M F 'PN(h(X)) for some bijection h : X ---4 N. And conversely, if M F 'P(h(X)) for some diagram-formula 'P(X) in C and an interpretation h : X ---4 M then there is a local submodel N of M with the universe heX) such that 'P = 'PN (up to a permutation of conjuncts).
Corollary 3.1. An .c-structure M is locally embeddable into a class K if and only if every diagram-formula realizable in M is realizable also in some structure from K. Lemma 3.8. For any class of .c-structures K the following holds:
8 Ucl(K) = L(K); 9 U cl(K) = SPu (K); 10 Dis(K) c(X), i.e, a subset p ~ cI>c(X) that can be realized in a structure from Mod(T). A type p is complete if it is a maximal type in cI> c(X) with respect to inclusion. It is easy to see that if p is a maximal type in X then for every formula i.p E cI> c( X) either i.p E P or --, i.p E p. Definition 4.1. A set p of atomic or negations of atomic formulas from cI> c(X) is called an atomic type in X relative to a theory T if pUT is consistent. A maximal atomic type in cI> c( X) with respect to inclusion termed a complete atomic type of T.
97 It is not hard to see that if p is a complete atomic type then for every atomic formula 'P E At.c(X) either 'P E P or -, 'P E p. Example 4.1. Let M be an .c-structure and m = (ml,"" m n ) E Mn. Then the set atpM(m) of atomic or negations of atomic formulas from .c (X) that are true in M under an interpretation Xi f---+ mi, i = 1, ... , n, is a complete atomic type relative to any theory T such that M E Mod(T).
We say that a complete atomic type p in variables X is realized in M = atpM(m) for some m E Mn. Every type p in T can be realized in some model of T (i.e., a structure from Mod(T)). If p cannot be realized in a structure M then we say that M omits p. There are deep results in model theory on how to construct models of T omitting a given type or a set of types. For an atomic type p ~ .c(X) by p+ and p- we denote, correspondingly, the set of all atomic and negations of atomic formulas in p. If S is a set of atomic formulas from .c(X) and M is an .cstructure then by VM(S) we denote the set {(ml, ... ,mn ) E Mn 1M 1= S(ml, .. " m n )} of all tuples in Mn that satisfy all the formulas from S. The set V M (S) is called the algebraic set defined by S in M. We refer to S as a system of equations in .c, and to elements of S - as equations in .c. Sometimes, to emphasize that formulas are from .c we call such equations (and systems of equations) coefficient-free equations, meanwhile, in the case when .c = .cA, we refer to such equations as equations with coefficients in algebra A. Following [4] we define Zariski topology on M n , n ~ 1, where algebraic sets form a prebasis of closed sets, i.e., closed sets in this topology are obtained from the algebraic sets by finite unions and (arbitrary) intersections. If p is an atomic type in .c in variables X = {Xl, ... , xn} then V M (p+) is an algebraic set in Mn. More generally, for an arbitrary type p in X by p+ we denote the set of all positive formulas in p, i.e., all formulas in the prenex form that do not have the negation symbol. If p is quantifier-free type, i.e., a type consisting of quantifier-free formulas, then formulas in p+ are conjunctions and disjunctions of atomic formulas. if p
Lemma 4.1. Let M be an .c-structure and n E N. Then for a subset ~ Mn the following conditions are equivalent:
V
• V is closed in the Zariski topology on Mn,• V = V M(P+) for some quantifier-free type p in variables {Xl, ... , x n }.
98
o
Proof. Straightforward. 4.2. Coordinate algebras and complete types
Let M be an .c-algebra. For a set 8 of atomic formulas from q,dX) denote by Rad M (8) the set of all atomic formulas from q,dX) that hold on every tuple from V M(8). In particular, if V M(8) = 0 then Rad M (8) = Atc(X). It is not hard to see that Rad M (8) is a congruent set of formulas, hence it defines a congruence that we denote by BRad(S). The .c-structure Tc(X)jBRad(S) is called the coordinate algebra of the algebraic set V M (8). If Y = V M(8) then the coordinate algebra TdX)jBRad(S) is denoted by feY) and Rad(8) - by Rad(Y). The following result gives a characterization of the coordinate algebras over an algebra M. Proposition 4.1. A finitely generated .c-algebra C is the coordinate algebra of some non-empty algebraic set over an .c-algebra M if and only if C is separated by M. Proof. Let Y be an algebraic set in Mn. With a point p = (ml' ... ,mn ) E M n we associate a homomorphism hp : Tc(X) ----; M defined by hp(t) = tM(ml, ... , m n ). Clearly,
BRad(Y)
=
n
ker hp-
pEY
Therefore, the diagonal homomorphism I1 pE Y : Tc(X) ----; a monomorphism f(Y)
= TdX)jBRad(Y)
I1PE Y M
induces
----; MIYI.
It follows that r(Y) E SP(M). Now, by Lemma 3.4 SP(M) = Res(M), so feY) E Res(M). Suppose now that C is a finitely generated .c-algebra from Res(M) with a finite generating set X = {Xl, ... , x n }. Let C = (X I 8) be a presentation of C by the generators X and relations 8 ~ Atc(X). In this case C is isomorphic to Tc(X)jBs. To prove that C is the coordinate algebra of some algebraic set over M it suffices to show that Rad M (8) = [8]. If (tl = t2) (j. [S] then there exists a homomorphism h : C ----; M with tj"1(h(XI), ... , h(xn)) i= t~(h(XI)' ... ' h(xn)). Obviously,
99
(h(XI),"" h(xn)) E V M(S) so (tl RadM(S) = [S].
=
t2) rf- RadM(S). This shows that D
Lemma 4.2. Let p be a complete atomic type in variables X. Then:
• p+ is a congruent set of formulas; • p+ = RadM(p+) for every £-structure M with V M(P)
=1=
0.
Proof. Indeed, since p is realized in some model M of T its positive part p+ satisfies the assumptions of Lemma 3.2, hence it is congruent. It follows that p+ determines a congruence ()p on Tc(X). Since p is complete one has p+ = RadM (p+). D Definition 4.2. Let X be a finite set of variables and p a complete atomic type in variables X. Then the factor-algebra Tc(X)/()p of the free £algebra Tc(X) is termed the algebra defined by the type p and the tuple (xI/()p, ... ,Xn/()p) is called a generic point of p.
Clearly, any complete atomic type p in variables X in a theory T is realized in the factor-algebra Tt:,(X)/()p at the generic point x (XI/()p,""xn/()p), so atpTcCX)/(lp(X)
=
p.
Indeed, for any atomic formula tl = t2, where tl, t2 E T,dX) one has (h = t2) E p if and only if h "'(lp t2, which is equivalent to the condition Tc(X)/()p F (h = t2) under the interpretation Xi I----' xi/()p' The generic point (xI/()p,"" xn/()p) satisfies the following universal property. If p is realized in some £-structure Mat (ml,"" m n ) E Mn then the map Xl -. ml,"" Xn -. mn extends to a homomorphism T.c(X)/()p -. M. Lemma 4.3. Let T be a universally axiomatized theory in.c. Then for any
finitely generated £-structure M the following conditions are equivalent:
1) M E Mod(T); 2) M = T.c(X)/()p for some complete atomic type p in T. Proof. Let X = {Xl, ... , xn} be a finite set and (X I S) a presentation of an £-structure M, i.e., M ~ Tc(X)/()s. If p = atpM(x), X = (Xl, ... , xn) then [S] = p+ and Tt:..(X)/()p ~ Tc(X)/()s ~ M. Therefore, 1) implies 2). To prove the converse, let p be an atomic type in T. We need to show that Tc(X)/()p E Mod(T). Since p is a type in T there exists a model N E Mod(T) and a tuple of elements fj = (YI,"" Yn) E Nn such that
100
=
N' is a substructure of N generated by Yl,'" ,Yn then N'. Since the theory T is axiomatized by a set of universal sentences one has N' E Mod(T). Hence, Tt:,(X)jBp E Mod(T). 0
p
atpN(y). If
T.c(X)jBp ~
4.3. Equationally Noetherian algebras The notion of equationally Noetherian groups was introduced in [4] and [7]. Let B be an algebra. For every natural number n we consider Zariski topology on Bn. A subset Y 1 then the amalgamated product
< a > *K < b, c; [b, c] =
1>
q
where K =< aP >=< b > is not CT. The situation for HNN extensions of CT groups is not as general but can be carried through for extensions of centralizers of abelian malnormal subgroups. Theorem 2.4. Let B be a CT gmup and K an abelian malnormal subgmup of B. Then the HNN extension
Bl
=< t,B;rel(B),r1kt = k for all k
E
K >
117
is aCT-group.
3. Commutative Transitivity, CSA and Universally Free Groups Myasnikov and Remeslennikov in their study of fully residually free groups introduced the concept of a CSA group (conjugately separated abelian group). First we recall the following necessary definition.
Definition 3.1. Let G be a group and H a subgroup of G. His malnormal in G or conj ugately separated in G provided g -1 H g n H = 1 unless g E H. Using this we define the concept of a CSA group.
Definition 3.2. A group G is a CSA-group or conjugately separated abelian group provided the maximal abelian subgroups are malnormal. Each CSA group must be CT. The converse however is not true in general.
Lemma 3.1. The class of CSA groups is a proper subclass of the class of CT groups. Proof. We first show that every CSA-group is commutative transitive. Let G be a group in which maximal abelian subgroups are malnormal and suppose that M1 and M2 are maximal abelian subgroups in G with z =J 1 lying in M1 n M 2. Could we have M1 =J M2? Suppose that w E M1 \ M 2. Then w- 1zw = z is a non-trivial element of w- 1M 2w n M2 so that w E M 2. This is impossible and therefore M1 c M2 . By maximality we then get M1 = M 2. Hence, G is commutative transitive whenever all maximal abelian subgroups are malnormal. We now show that there do exist CT groups that are not eSA. In any non-abelian CSA-group the only abelian normal subgroup is the trivial subgroup 1. To see this suppose that N is any normal abelian subgroup of the non-abelian CSA-group G. Then N is contained in a maximal abelian subgroup M . Let g ~ M . Then N
= g-lNgn N c g-lMg n M.
The fact N =J 1 would imply that gEM which is a contradiction. Now let p and q be distinct primes with p a divisor of q - 1. Let G be the non-abelian group of order pq . Then it is not difficult to prove that
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the centralizer of every non-trivial element of G is cyclic of order either p or q . Thus G is commutative transitive. However, the (necessarily unique) Sylow q-subgroup of G is normal in G. Hence from the argument above G cannot be CSA. 0 Although the class of CSA groups is a proper subclass of the CT groups in the presence of full residual freeness (in fact even in the presence of just residual freeness) they are equivalent. Definition 3.3. A group G is residually free if for each non-trivial 9 E G there is a free group Fg and an epimorphism hg : G ---; Fg such that hg(g) #- 1. Equivalently for each 9 E G there is a normal subgroup N g such that G/Ng is free and 9 rJ. N g • The group G is fully residually free provided to every finite set S c G \ {1} of non-trivial elements of G there is a free group Fs and an epimorphism hs : G ---; Fs such that hs(g) #- 1 for all 9 E S. Lemma 3.2. If G is fully residually free then GT
== GSA.
The proof of this depends on a beautiful theorm due independently to Gaglione and Spellman [GS] and Remeslennikov [R] tying together full residual freeness and the property of being universally free which we will explain shortly (see [R] for a proof of Lemma 2.3). In the 1960's G. Baumslag [GB 1] proved that a surface group is residually free, answering a question of Magnus. To do this he introduced what is now called extensions of centralizers. This concept became one of the main tools used by Kharlampovich,Mayasnikov and Remeslennikov in their structure theory of fully residually free groups and by Kharlampovich and Maysnikov in their solution to the Tarski problems. As somewhat of an offshoot of this B. Baumslag [BB 1] proved: Theorem 3.1. If G is residually free then the following are equivalent: (1) G is fully residually free (2) G is GT From this a truly amazing result developed that is in some sense the beginning of the solution of the Tarski problem. This was done independently by Gaglione and Spellman and Remeslennikov: Recall first some ideas from logic and model theory. We start with a first-order language appropriate for group theory. This language which we denote by Lo is the first-order language with equality
119
containing a binary operation symbol . a unary operation symbol -1 and a constant symbol 1. A sentence in this language is a logical expression containing a string of variables x = (Xl, ... , x n ), the logical connectives V, /\, rv and the quantifiers V, 3. A universal sentence of Lo is one of the form \fX{ ¢(x)} where x is a tuple of distinct variables, ¢(x) is a formula of Lo containing no quantifiers and containing at most the variables of X. Similarly an existential sentence is one of the form 3x{ ¢(x)} where x and ¢(x) are as above. If G is a group then the universal theory of G consists of the set of all universal sentences of Lo true in G. We denote the universal theory of a group G by T hv( G). Since any universal sentence is equivalent to the negation of an existential sentence it follows that two groups have the same universal theory if and only if they have the same existential theory. The set of all sentences of Lo true in G is called the first-order theory or the elementary theory of G. We denote this by Th(G). We note that being first-order or elementary means that in the intended interpretation of any formula or sentence all of the variables (free or bound) are assumed to take on as values only individual group elements - never, for example, subsets of, nor functions on, the group in which they are interpreted. The Tarski conjectures, solved independently by Kharlampovich and Myasnikov (see [KHM 1-5]) and Sela (see [Se 1-5]), say essentially that all countable nonabelian free groups have the same elementary theory. The following was well-known and much simpler. Theorem 3.2. All nonabelian free groups have the same universal theory. As we will see all finitely generated fully residually free groups will also have the same universal theory as the class of nonabelian free groups. Definition 3.4. A universally free group G is a group that has the same universal theory as a nonabelian free group. Gaglione and Spellman [GS] and independently Remeslennikov [Re] were able to extend the theorem of B. Baumslag to show that fully residually free is equivalent to universally free and that these (in the presence of residual freeness) are equivalent to both being CT and being CSA. As mentioned this is in some ssnse the starting off point for the solution of the Tarksi conjectures. Theorem 3.3 (GS). ,[Re) If G is residually free then the following are equivalent:
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(1) G is fully residually free
(2) G is GT (3) G is GSA (4) G is universally free Since this result a complete structure theory and algorithmic theory of the fully residually free groups has been developed. An important aspect of this development is that elements of fully residually free groups can be expressed as infinite words on a generating system. These infinite words can be manipulated and handled in an analogous manner to ordinary words in free groups (see [KMRS]). Commutative transitivity becomes essential in building examples of fully residually free groups classifying them via the following construction. Definition 3.5. Let G be a CT group, let u E G\ {I} and let M = Za(u) where Za(u) is the centralizer of u in G. Suppose A is an abelian group. Then the group H
=
is cyclic then H =
z, for all z EM>
and is called the free rank one extension of the centralizer M of u in
G. Theorem 3.4. (Baumslag, Myasnikov, Remeslennikov [BMR3j) Let G be a fully residually free group and A an abelian fully residually free group. Then a centralizer extension of G by A is again fully residually free. The proof of this result which is fundamental in all further considerations of fully residually free groups depends on the fact that the result can be reduced to free rank one extensions of centralizers and then on the following "big powers" argument. It is not hard to see that in a free group F if botn'bl ... tnkbk = 1 for infinitely many values of nl, infinitely many values of n2"" infinitely many values of nk then t must commute with at least one of bo , ... , bk . Hence the family of homomorphisms ¢k : F(u, t) ----t F from the rank one extension of the centralizer GF (u) into F, defined for every positive k by ¢(t) = uk and ¢k IF= id, is a discriminating family, as required.
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As mentioned earlier, G.Baumslag [GB 1] used this type of argument to show that the orient able surface groups 8 g with g 2: 2 are all residually free. This answered a question posed by Magnus. Recall that for g 2: 2 the group 8 g which is the fundamental group of an orient able surface of genus g has the presentation
8 g =< aI, bl , ... , ag, bgi [aI, bl] ... [a g, bg] = 1 > . Baumslag observed that each 8 g embeds in 8 2 and residual freeness is inherited by subgroups so it suffices to show that 82 is residually free. He actually showed more. If F is a nonabelian free group and u E F is a nontrivial element which is neither primitive nor a proper power then the group K given by
=< F * Fi U
K
= U
>
where F is an identical copy of F and u is the corresponding element to u in F, is residually free. A one-relator group of this form is called a Baumslag double. In our terminology he proceeded by embedding K in the free rank one extension of centralizers H
=
by K
=
.
The group H is then residually free and hence K is residually free. Therefore every Baumslag double is residually free. The group
8 2 =< al,b l ,a2,b 2i[al,b l ] = [a2,b 2] > is a Baumslag double answering the original question. The class of finitely generated fully residually free groups was introduced in a different direction by Sela in his proof of the Tarski problems. In Sela's approach these groups appear as limits of homomorphisms of a group G into a free group. In this guise they are called limit groups. Therefore a limit group is a finitely generated fully residually free group. The paper by Bestvinna and Feign [BeF] gives a nice description of the equivalence of the two approaches.
4. The Commutative Transitive Kernel Commutative transitivity and universal freeness raised certain questions about the relationship between fully residually free and n-free.
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Definition 4.1. A group G is n-free if any subset of n or fewer elements in G generate a free group. Observe that if G is an n-free group then G is also an m-free group for all 1 :s; m < n. The I-free groups are precisely the torsion free groups. 2-free groups are commutative transitive. Lemma 4.1. If G is an n-free group for any n transitive.
~
2 then G is commutative
Proof. Suppose that G -I- 1 is a 2-free group and let u E G. Let M = Ga (u) . We claim that M is locally cyclic and therefore abelian. Suppose a, b E M. Since G is 2-free, a, u generate a free group. Since a, u commute this must be cyclic and hence they are both powers of a single element g. Thus a = gCt, U = g{3 for some Ct, (3. Similarly, since band u commute and b, u generate a free group, we have an element h with b = h ii , U = h'Y for some 8, 'Y. Now consider < h, g >c G. This is free because G is 2-free however this has the relation g{3 = h'Y. Therefore < h, 9 > must be cyclic and hand 9 are both powers of a single element. Hence a and b are both powers of this element and any 2-generator subgroup of M is cyclic. A straightforward induction then shows that any n-generator subgroup < aI, ... , an >C Mis also cyclic. Thus, centralizers of non-trivial elements are locally cyclic and 2-free groups are commutative transitive as claimed. 0 The concept of n-freeness arises in many places. From straightforward topological considerations it follows that an orientable surface group of genus 9 ~ 1 is 2g - 1 free while a nonorientable surface gorup of genus 9 ~ 2 is g - 1 free. These results were generalized in an algebraic manner by Fine, Gaglione,Rosenberger and Spellman (see [FGRS 1]). In order to further study commutative transitivity, Fine, Gaglione, Rosenberger and Spellman [FGRS 2] introduced a subgroup T( G) called the commutative transitive kernel that carries the information about commutative transitivity in much the same way that the derived group G' carries the information about commutativity. Definition 4.2. Let N be a normal subgroup of G and let To(G, N) = N and inductively define Tn+I(G, N) = subgroup of G generated by Tn(G, N) tgether with those commutators [a, c] such that aRnc where Rn is the transitive closure of the relation on G - Tn(G, N) by xy == yx mod Tn(G, N). Now let
123
Then T(G,N) eNG' and T(G,N) is normal in G Lemma 4.2. Let N be normal in G. Then GjN is CT if and only if N = T(G, N). Further GjT(G, N) is CT Definition 4.3. The commutative transitive kernel of a group G, denoted T(G), is T(G, 1). The commutative transitive kernel behaves in much the same way as the derived group. In particular we have the following theorem. The important result is Theorem 4.1. Let G be a group. Then (1) T( G) is a characteristic subgroup of G and contained in G' (2) GjT(G) is CT (3) Gis CT if and only ifT(G) = 1 Unfortunately the commutative transitive kernel is not canonical for commutative transitivity in the same sense as the derived group is for commutativity. To make this precise let F be a covariant functor on the category of groups. We say that F is a T-like functor if for each group G [lJ F(G) is a characteristic subgroup f G and is contained in G' [2J G j F( G) is commutative transitive [3J G is commutative transitive if and only if F(G) = {I}. The set of T-like functors can be partially ordered by Fl :::; F2 if Fl (G) c F2(G) for every group G. A canonical subgroup for commutative transitivity would be equivalent to a minimum T-like functor. In [FGRS 1J the following was proved. Theorem 4.2. The family of T-like functors has no minimum. C. Delizia and C. Nicotera further studied the commutative transitive kernel for locally finite groups [DN 1,2J. They also developed an analog of the commutative transitive kernel for power commutative groups [DN 3,4J and study the structure of this kernel for locally nilpotent groups. In another direction they consider groups where nilpotency is transitive on 2-generator subgroups [DN 5J.
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5. RG Groups and a Classification of One-Relator CT Groups In presenting examples of infinite CT groups we mentioned that one-relator groups with torsion are CT. The question then arises as to how to classify both the one-relator CT groups and the one-relator CSA groups. In this section we describe a recent classification of such one-relator groups done by Fine, grose Rebel, Myasnikov and Rosenberger [FgRMR]. This classification is related to two more general questions. (1) The general classification of one-relator fully residually free groups (2) The Gersten conjecture that a torsion-free one-relator group is hyperbolic if and only if it does not contain any Baumslag-Solitar group B m,n =< x, y; yx m y -1 = x n >, mn r--I- 0 . Recall that one-relator groups with torsion are hyperbolic. In order to present the classification we need another concept introduced by Fine and Rosenberger [FR] and independently by M.Cohen and M.Lustig [CL] Definition 5.1. A group G is a restricted Gromov or RG group if given < g, h > is cyclic or there exists a positive integer t with gf -1= 1,h t -1= 1 and < gt,ht >=< gf > * < ht >. g, h E G either
In particular free groups and torsion-free hyperbolic groups are RG. Theorem 5.1 (FR). Torsion-free RG groups are CT. Now the classification. For one-relator groups with torsion, combining results of Fine and Rosenberger [FR] and Gildenhuys,Kharlampovich and Myasnikov [GKM] we obtain the following result ([FgRMR]). Theorem 5.2. ((FgRMRJ) Let G be a one-relator group with torsion. Then the following are equivalent: (1) Gis CSA (2) Gis RG (3) G does not contain a copy of the infinite dihedral group < x, y; x2 = y2 = 1 > For torsion-free one-relator groups the situation is different. A torsionfree one-relator group fails to be a CSA group if and only if its contains
125
a copy of some nonabelian Baumslag-Solitar group BI,n with n -I- 1 or a copy of the group FI x Z, the direct product of a free group of rank 2 and an infinite cyclic group (see [FgRMR]). In [FgRMR] it was proved that a one-relator group fails to be an RG-group if and only if it contains a copy of one of the Baumslag-Solitar groups BI,n. Recall that BI,1 is a free abelain group of rank 2. It follows that if G is a torsion-free one-relator group which does not contain a free abelian subgroup of rank 2 then the following are equivalent. (1) G is a CSA group (2) G is an RG-group (3) G does not contain a copy of some BI,n with n -I- -1,0,1. Further if G is a torsion-free one-relator group then G is CT if and only if it does not contain a copy of F2 x Z or a copy of the Kleian bottle group BI,-l. Combining all these we get the following results. Theorem 5.3. ([FgRMRj) Let G be a torsion-free one-relator group which does not contain a copy of Z x Z = BI,I. Then the following are equivalent: (1) G is GSA (2) Gis RG (3) G does not contain a copy of one of the Baumslag-Solitar group BI,m =< x, y : yxy-l = xm > with mE Z \ {-I, 0, I}. Theorem 5.4. ([FgRMRj) Let G be a torsion-free one-relator group. Then G is GT if and only if G does not contain a copy of F2 X Z or a copy of the Baumslag-Solitar group BI,-l =< x, y; yxy-l = X-I > (the Klein-bottle group). Notice that since one-relator groups with torsion are commutative transitive, this fact together with Theorem 4.3 provides the total classification of one-relator commutative transitive groups. 6. Commutative Transitivity and Discriminating Groups As an outgrowth of the development of the theory of algebraic geometry over groups (see [BMR2] [BMR1]), Baumslag, Myasnikov and Remeslennikov introduced the concept of discriminating groups. This class of groups developed a life of their own and have been extensively studied (see [FGMRS] and the references there). Definition 6.1. Let G and H be groups. G separates H provided that to every nontrivial element h of H there is a homomorphism 'Ph : H -+ G
126
such that CPh(h) -=I- 1. G discriminates H if to every finite nonempty set S of nontrivial elements of H there is a homomorphism CPS : H ----+ G such that cps(s) -=I- 1 for all s E S. The group G is discriminating provided that it discriminates every group it separates. Algebraic geometry over groups was created as a tool to attack the celebrated Tarski conjectures on the elementary theory of free groups. By analogy with classical algebraic geometry we may view the discrimination of H by G as an approximation to H much like the localization of a ring at a prime. (Think of a set of generators for H as a set of variables.) The following is the main criterion for determining whether a group is discriminating. Lemma 6.1. ([BMRj) A gmup G is discriminating if and only if G discriminates G x G. In the application of algebraic geometry it is as important to know both whether a group is discriminating and whether it is not discrimninating. Hence there are general questions on nondiscrimination of various classes of groups. General idea to show nondiscrimination is to show that some universal property which is true in G but cannot be true in G x G or to find a number - dimension etc. - which is additive so that this number cannot hold in both G x G and in G. Here commutative transitivity plays a large role. Lemma 6.2. A nonabelian CT gmup is nondiscriminating. It follows that all the following classes of groups are nondiscriminating: (1) Any torsion-free hyperbolic group and in particular any nonabelian free group is non discriminating (2) Nonabelian free solvable groups and their nonabelian subgroups are nondiscriminating. (3) The free product of two nondiscriminating groups is nondiscriminating. In subsequent work it was shown that a nilpotent group is discriminating if and only if its torsion-free abelian (see [FGMS]). 7. An Extension of Commutative Thansitivity In [BFGRS] a study was initiated to extend the concept of commutative transitivity. This begins in the following direction.
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Definition 7.1. Let V be a variety of groups that contains the abelian variety A. Then we say that a group G is a centralizer- V group or evgroup if for each 9 E G the centralizer of 9 is in V.
In this context CT-groups are centralizer abelian or CA - groups. Since varieties are closed under subgroups is is clear that any group G in V is a eV-group. Further since we are assuming that the variety V contains the abelian variety it follows that any CT-group is ev. There are certainly eV-groups that are not CT-groups. For example the group G
=< a, b; aba- 1 = b- 1 >
is not CT but is eM where M is a variety which contains the abelian and metabelian variety. We now show that there are nontrivial examples of eV-groups, that is groups G that are not in V or commutative transitive but which are ev. We call a group a nontrivial eV-group if it is ev and not in V and not CT.
Theorem 7.1. A free product of nontrivial eV-groups is a nontrivial evgroup. Theorem 7.2. Let G and H be nontrivial eV-groups. Then the unrestricted wreath product Gwr H is a nontrivial ev -group.
Mimicing methods of Levin and Rosenberger [LR] We can show that the class of eV-groups is closed under certain types of group amalgams.
Theorem 7.3. Let A and B be eV-groups and An B = K proper and malnormal in both A and B. Then the amalgamated free product A *K B is a eV-group. In particular a nontrivial free product of eV-groups is again ev. Theorem 7.4. Let B be a eV-group and K a nontrivial abelian malnormal subgroup of B. Then the HNN group
Bl =< t,B;C1kt = k for all k E K > is a eV-group.
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8. References [AAR] P. Ackermann, V. grosse Rebel and G. Rosenberger, On Power and Commutation Transitive, Power Commutative and Restricted Gromov Groups Cont. Math. ,360,2004,1-4 [GB 1] G. Baumslag On generalized free products Math. Z., 78, 1962,423438 [BFGS] G.Baumslag, B. Fine, A.M. Gaglione and D. Spellman, Reflections on discriminating groups, J. Group Theory, 10, 2007, 87-99 [BFGRS] G.Baumslag, B. Fine, A.M. Gaglione,G.Rosenberger and D. Spellman, On Centralizer Varietal Groups in preparation [BMR1] G. Baumslag, A.G. Myasnikov, V.N. Remeslennikov, Algebraic geometry over groups I. Algebraic sets and ideal theory J. Algebra, 219, 1999, 16-79 [BMR2] G. Baumslag, A.G. Myasnikov, V.N. Remeslennikov, Discriminating and co-discriminating groups J. of Group Theory, 3, 2000,467-47 [CL] M. Cohen and M. Lustig, Very small group actions on JR-trees and Dehn twist automorphisms Topology, 34, 1995, 575-617 [DN 1] C. Delizia and C.Nicotera, On the Commutative Transitive Kernel of Locally Finite Groups Alg. Colloquium, 10, 2003, 567-570 [DN 2] C. Delizia and C.Nicotera, On the Commutative Transitive Kernel of Certain Infinite Groups JP Journal of Algebra, Number Theory and Applications, 5, 2005, 421-427 [DN 3] C. Delizia and C.Nicotera, On the Power-Commutative Kernel of Locally Nilpotent Groups Int. J. of Math. and Math. Sciences, 17, 2005, 2719-2722 [DN 4] C. Delizia and C.Nicotera, On the Power-Commutative Kernel of Finite Groups Journal of Science, Ferdowsi University of Mashad , 5, 2005, 3-8 [DN 5] C. Delizia and C.Nicotera, Groups in Which the Bounded Nilpotency of Two-Generator Subgroups is Transitive preprint [F] B.Fine, On Power Conjugacy and SQ-universality for Fuchsian and Kleinian Groups, in Modular Functions in Analysis and Number Theory, University of Pittsburgh Press, 1983, 41-55 [FGMS] B. Fine, A.M. Gaglione, A.G. Myasnikov and D. Spellman, Discriminating groups: A Comprehensive Overview CRM Preprint , 2006
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[FGMS1] B. Fine, A.M. Gaglione, A.G. Myasnikov and D. Spellman, Discriminating groups J. of Group Theory, 4, 2001, 463-474 [FGRS 1] B.Fine,A.M.Gaglione,G.Rosenberger and D. Spellman, The Commutative Transitive Kernel Algebra Colloquium, 2, 1997, 141-152 [FGRS 2] B.Fine,A.M.Gaglione,G.Rosenberger and D. Spellman, Free Groups and Questions About Universally Free Groups, in Proceedings of Groups St. Andrews/Galway 1993, London Mathematical Soc. Lecture Notes Series 211, 1993, 191-204 [FR 1] B.Fine and G.Rosenberger, On Restricted Gromov Groups Comm.in Aig. , 20, 8, 1992, 2171-2182 [FgRMR] B.Fine, V. grosse Rebel, A. Myasnikov and G. Rosenberger, A Classification of CSA, Commutative Transitive and Restricted Gromov One-Relator Groups Result. Math. to appear [GS] A.M. Gaglione and D. Spellman, Even More Model Theory of Free Groups, in Infinite Groups and Group Rings, World Scientific, 1993, 37-40 [GKM] D.Gildenhuys, O. Kharlampovich and A. Myasnikov, CSA Groups and Separated Free Constructions Bull. Austral. Math. Soc. ,52, 1995, 63-84
[K] M.Kasabov On Discriminating Solvable Groups preprint [LR] F. Levin and G. Rosenberger, On Power Commutative and Commutation Transitive Groups, in Proc. Groups St Andrews 1985 , Cambridge University Press, 1986 , 249-253 [LS] R.C.Lyndon and P.E. Schupp, Combinatorial Group Theory, Springer-Verlag 1977 [MKS] W. Magnus, A. Karrass, and D. Solitar, Combinatorial Group Theory, Wiley 1966, Second Edition, Dover Publications, New York 1976 [MRe] A.G. Myasnikov and V.N. Remeslennikov Exponential groups 1: Foundations of the theory and tensor completion Omsk University, N9 1993 1-25 [MRe 1] A.G. Myasnikov and V.N. Remeslennikov, Exponential groups, preprint New York, 1994 [MRe 2] A.G. Myasnikov and V.N. Remeslennikov, Exponential groups 2: Extensions of centralizers and tensor completion of CSA groups, preprint [MS] A.M. Myasnikov and P. Shumyatsky, On Discriminating Groups and c-Dimension J. of Group Theory, 200
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Bull. Amer.
[P] S.J. Pride, The two generator subgroups of one-relator groups with torsion Trans. Amer. Math. Soc. , 234, 1977, 483-496 [Su] M. Suzuki, The Nonexistence of a Certain Type of Simple Groups of Odd Order Proc. Amer. Math. Soc. ,8,1925,686-695 [W] L.Weisner, Groups in which the normalizer of every element but the identity is abelian Bull. Amer. Math. Soc. , 31, 1925, 413-416 [Wu] Y.F. Wu Groups in which Commutativity is a Transitive Relation J. of Algebra, 207, 1998, 165-181
GROUPS UNIVERSALLY EQUIVALENT TO FREE BURNSIDE GROUPS OF PRIME EXPONENT AND A QUESTION OF PHILIP HALL A. Gaglione
Department of Mathematics U.S. Naval Academy Annapolis, MD 21402 S. Lipschutz
Department of Mathematics Temple University Philadelphia, PA 19122 D. Spellman
Department of Mathematics Temple University Philadelphia, PA 19122
Subject Classification: Primary 20E26; Secondary 20E05, 20E06 Abstract :This paper proposes the Tarski Problem for free groups in a Burnside variety, B n , where n is a sufficiently large odd integer so that Adian's results hold. We note that just as in the case of absolutely free groups it is easy to show that the nonabelian free groups in Bn for n as above are all universlly equivalent.
1. Introduction
We bear in mind two problems which resisted solution for decades and succumbed only to the fresh attacks of supremely talented mathematicians. For our purposes we shall dub the questions (1) and (2) below as The Tarski Problem and The Burnside Problem respectively. (1) Are the nonabelian free groups elementarily equivalent? (2) Must the finitely generated nonabelian free Burnside groups of fixed finite exponent be finite? 131
132
Of course the answers to (1) and (2) are now known to be (1) yes and (2) no respectively. (1) was solved independently by Kharlampovich and Myasnikov on the one hand and by Sela on the other. Kharlampovich and Myasnikov applied algebraic geometry over groups invented by G. Baumslag, Myasnikov and Remeslennikov while Sela invented Diophantine geometry in groups to solve the Tarski problem. Let's focus on algebraic geometry over groups. By way of analogy let us consider a Noetherian integral domain R. The closed subsets in the Zariski topology on affine n-space over R are precisely the affine algebraic subsets. In order for the analog of this scenario to go through in groups the proper notions of domain and Noetherian had to be defined. One consequence of the correct definition of domain is that every nonabelian eSA group (See Section 3) is a domain. Since free groups are eSA, nonabelian free groups are domains. The correct notion of Noetherian is equationally Noetherian (See Section 6). Happily free groups are also equationally Noetherian. One felicitous consequence of this confluence of facts it, the existence and uniqueness (in the usual sense) of the decomposition of a closed set into a finite union of irreducible affine algebraic subsets. (We need to reserve the word variety for an equational class in this paper.) While the groups free in the variety of all groups are elementarily equivalent (provided their rank exceeds 1) it is easy to see that the corresponding result is false for groups free in many other varieties. For example, one can distinguish the free nilpotent groups (of fixed class) of different finite ranks by first order sentences. Perhaps the variety of all groups is unique in this regard? Or is it! We now turn our attention to Question (2). A negative answer was first provided by Adian who showed that, for all sufficiently large odd n, the nonabelian free groups in the variety Bn determined by the law xn = 1 were all infinite. Soon afterward Sirvanjan proved that, just as in the variety of all groups, the free group of countably infinite rank in the variety Bn embeds in its group free of rank 2 for all sufficiently large odd n. From this it easily follows that the nonabelian free groups in the varieties Bn, for all sufficiently large fixed odd n, have the same universal theory in the sense of first order logic. For our purposes we can say the most when n = p is a sufficiently large prime. In any event, if n is a sufficiently large odd integer, the free groups in Bn are eSA; hence, the nonabelian free groups in such varieties are domains. We do not know whether or not they are equationally Noetherian. The Tarski problem relativized to such Bn will require new techniques for its solution. This paper provides some halting first steps and should be viewed as an invitation to our colleagues to ponder
133
possible approaches.
2. Preliminaries We let w be the first limit ordinal, which we identify with the first infinite cardinal ~a. If Hand G are groups we say that G is H -inclusive provided it contains a subgroup isomorphic to H; G is H -exclusive provided it is not H-inclusive. For each positive integer n, shall be a group cyclic of order n. If G is a group and H is a nonempty class of groups, then we say that H separates G provided for every 9 E G\ {I} there is a group Hg E H and a homomorphism Hg such that G, (g, h) f---+ gh. A universal sentence of La is one of the form 'v'x 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.
137
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} 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
=' 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
153
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
154
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 = (~).
155
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
156
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.
157
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.
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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.
.
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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 ... .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
. For each 1 : 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 >] = [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 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)
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*(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
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):