Abstract SOME REMARKS ON ALMOST FINITELY GENERATED NILPOTENT GROUPS

SOME REMARKS ON ALMOST FINITELY GENERATED NILPOTENT GROUPS PETER HILTON AND ROBERT MILITELLO Dedicated to the memory of Pere Menal We identify two generalizations ofthe notion ofa finitely generated nilpotent. Thus a nilpotent group G is fgp if Gp is fg as p-local group for each p; and G is fg-like if there exists a fg nilpotent group H such that Gp = Hp for all p. Then we have proper set-inclusions {fg} C {fg-like} C {fgp}. We examine the extent to which fg-hke nilpotent groups satisfy the axioms for a Serre class . We obtain a complete answer only in the case that [G, G] is finite . (The collection of fgp nilpotent groups is known te form a Serre class in the extended sense) .

For if B is the subgroup of Q generated by the rationals (P, all p), then B is not fg but is plainly 71-like ; and if A = ® 7L/p then A is plainly P fgp, but it cannot be fg-like since the torsion subgroup of any fg-like group must be finite .
Moreover, this example shows that the fg-like groups do not form a Serre class, since we have a short exact sequence.7L >--r B -» A.
However, for abelian groups, the only axiom which fails is that which asserts that a quotient group óf a member of the class is again a member of the class .This we show in the next section, where we regard the Serre axioms relating te a short exact sequence as the principal axioms and the remaining axioms as subsidiary.However, the fact that a subgroup of an abelian fg-like group is fg-like, and the fact that an abelian extension of an fg-like group by an fg-like group is fg-like, both follow from an easy characterization of abelian fg-like groups as those abelian fgp groups whose torsion subgroups are finite.For nilpotent groups we do not know if this characterization holds ; certainly fg-like nilpotent groups have finite torsion subgroups, but we do not know whether fgp nilpotent groups with finite torsion subgroups are necessarily fg-like .If the characterization held then the corresponding Serre axioms for fg-like nilpotent groups could be proved just as easily as in the abelian case.
In Section 3 we go as far as we can in the nilpotent case.Of course, the homology axiom holds; for the homology groups of a fg nilpotent group are fg, and, if G is K-like, with K fg, then Hk (G) is Hk (K)-like .As to the principal axioms, one of couse is falle and, as to the others, we are only able to prove them in the case of those groups G such that [G, G] is finite .This we do by means of our main theorem, Theorem 3.1, which asserts that if Q = GIN with N finite, then G is fg-hke if and only if Q is fg-like .From this it readily follows (Theorem 3.5) that if G is fgp with TG finite and if [G, G] is finite, then G is fg-like .
In an Appendix we show that Schur's Theorem, namely, that if G is a group with G/ZG finite, where ZG is the center of G, then [G, G] is finite, admits a converse if G is nilpotent fgp.Moreover, precisely the same primes enter into the orders of G/ZG and [G, G] .

. The Abelian case
We have already observed that a quotient of an fg-like abelian group need not be fg-hke .We will show in this section that this is the only axiom for a Serre class which fails.As to the subsidiary axioms, this follows from the following composite theorem.

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Theorem 2.1 .Let A be M-like and B be N-like (where all groups are abelian).Then AS to the principal axioms, we have only to prove (a) that a subgroup of an fg-like abelian group is fg-hke ; and (b) that an abelian extension of an fg-hke group by an fg-like group is fg-hke.We base these re- sults on the following obvious characterization of abelian fg-like groups .(Compare Theorem 1.3 of [CH2]) .
Proposition 2.2.Let A be an abelian group .Then A is fg-like if and only if A is fgp and TA is finite .
Proof. .If A is M-like with 111 fg, then TA = TM which is evidently finite .Thus if A is fg-like, then A is fgp and TA is finite.Conversely, suppose that TA is finite and A is fgp .Then FA is 7L k-like, for some k, so that (see Theorem 1.1 of [CH2]) Ext(FA,TA) = 0 and A = TA®FA .It follows that A is M-like, where M = TA ® 7Lk .
Corollary 2 .3.(a) A subgroup of an fg-like abelian group is fg-like .
(b) An abelian extension of an fg-like group by an fg-like group is fg-like.
Proof.. Since fgp abelian groups form a Serre class [H2], it remains only, in the light of Proposition 2.2, to show that the property of having a finite torsion subgroup is preserved by subgroups and extensions .This is absolutely clear for subgroups .For extensions, we observe that, if is exact, so is G'-G >G" TG' >-+ TG -» 7r(TG), ® Np = and 7r(TG) C TG" .Thus, if TG' and TG" are finito, so is TG.
Remark.We notice that the proof of Corollary 2.3 works in the nilpotent case, provided the characterization of Proposition 2.2 remains valid for nilpotent groups .Certainly fg-hke nilpotent groups are fgp with finite torsion subgroups, but we have not been able to establish the converse .Thus we have also not succeeded in generalizing Corollary 2.3 to nilpotent groups .

The Nilpotent case
Our basic result in the nilpotent case is the following.Theorem 3.1 .Let N >--> G -» Q be a short exact sequence of nilpotent groups with N finite .Then G is fg-like if and only if Q is fg-like .
Proof. .Suppose G is K-like, with K fg.We may then assume that TG = TK and that, `dp, Gp = Kp .Now N C TG = TK C K and Np is normal in Kp(= Gp) for all p.It thus follows from [HM] that N is normal in K. Set L = K/N, so that L is fg .For all p, we have Np -Gp -Qp 1 1 1 1 5 Np '--' Kp -+' Lp yiclding isomorphisms Qp -Lp, so that Q is L-like .Conversely, suppose that Q is L-like, with L fg; and suppose further that N is commutative.Then the sequence N >--> G -» Q determines a nilpotent action of Q on N, and then the extension represents an element 97 = [G] E H2 (Q; N), where N is regarded as a (nilpotent) Q-module .Now there are associated nilpotent actions of Qp on Np, for all p, such that The isomorphisms (3.1), (3 .2) compose to an isomorphism H2 (Q ; N) H2 (L ; N), under which 77 corresponds to, say, ~, where ( is represented by the sequence N >-+ K -» L .Then K is nilpotent and certainly fg.Moreover, for each p, we have an isomorphism showing that G is K-like .We now complete the converse argument by induction on nil N.For the sequence N >--> G -» Q gives rise to a sequence Since [N, N] is finite with nil [N, N] < nil N, our inductive hypothesis allows us to infer that G is fg-like . Remark.The abelian version of Theorem 3 .1 is trivial .For then Corollary 2.3(b) assures us that G is fg-like if Q is fg-like ; and, in the other direction, the finiteness of N tells us that TG maps onto TQ, so Q is fg-like by Proposition 2.2.Corollary 3.2 .Let G be a nilpotent group with finite commutatorsubgroup .Then G is fg-like if and only if Ga b is fg-like .
Proof.Of course we need no assumption on [G, G] to infer that Ga b iS fg-hke if G is fg-like ; for if G is K-like, then Ga b is Kab-like .However, in the other direction, we must apply Theorem 3.1 .Theorem 3.3.Let G be fg-like and let H C_ G be a subgroup such that Hf1 [G, G] is finite .Then H is fg-like .In particular, all subgroups of G are fg-like if [G, G] is finite .

Appendix : Groups with finite commutatorsubgroup
A famous theorem due to I. Schur asserts that if G/ZG is finite, then [G, G] is finite .A study of its proof (we give a homological proof below) shows that more is true.Given any finite group N, let -r(N) be the set of primes dividing the order of N. Then we may conclude that 7-[G, G] C_ r(GIZG) .In the context of nilpotent groups, we have a stronger result.Proof.We first prove Schur's Theorem .Thus if G/ZG is finite then G'/G' n ZG is finite, where G' = [G, G], and r(G'/G' n ZG) C_ -r(G/ZG).Thus it remains to prove that G' n ZG is finite, with -r(G' n ZG) C_ T(G/ZG) .Now H2(G/ZG) is finite with T(H2 (G/ZG)) C_ r(GIZG) .Moreover there is an exact sequence H2(G/ZG) > ZG -> Gab, which shows that G' n ZG is a homomorphic image of H2(G/ZG) .We conclude that G' n ZG is finite with r(G' n ZG) C_ T(G/ZG), as required.
Indeed, we have shown that T[G, G] C T(G/ZG), as claimed .
Conversely, suppose that G is an fgp nilpotent group with G' finite.Set T = -r(G') .Then, for all g, h E G, [g, h]' = 1 for some T-number n.If b = [g, h], we have gb = h-I gh so that, by Corollary 6.2 of [H1], gn`= (h-lgh)n' = h _I g nc h, where ni1G <_ e.Thus 9n`E ZG, so that G/ZG is a torsion group with bounded exponent dividing n' .But G is fgp, so G/ZG is fgp .It follows that each (G/ZG) p is finite and (G/ZG)p is trivial unless p1n°.We conclude that G/ZG is a finite group with T(G/ZG) C T = T [G, G] .This completes the proof of the theorem.
this isomorphism, rl corresponds to {77p}, where qp E H2 (Q p; Np) is represented byN p >--> Gp -Qp .We may, as before, assume that Lp = Qp and then we let L act on Np via ep : L -+ Lp .This action is, of course, nilpotent, and so is the induced action of L on N = jjNp .Thus we obtain, paralleling (3N) ^--' 11 H2 (L p ; Np)(= fl H2 (Qp ; Np)) p p Theorem 4.1.Let G be an fgp nilpotent group.Then G/ZG is finite if and only if [G, G] is finite and then T(G/ZG) = T[G, G] .