LOCALLY SOLUBLE GROUP S WITH ALL NONTRIVIAL NORMAL SUBGROUPS ISOMORPHIC

In paper [2] we considered groups which are isomorphic to all of thei r nontrivial normal subgroups . The question as to which infinite group s have this property P, say, was first raised by Philip Hall . It was shown in [2] that, if G is a finitely generated infinite P-group which contains a proper normal subgroup of finite index, then G is cyclic, and our conjecture is that Z is the only finitely generated infinite P-group which is not simple . It was further remarked in [2] that is should perhaps be possible to deal with the locally soluble case, and this note represents a step in this direction . The following is proved .

In paper [2] we considered groups which are isomorphic to all of thei r nontrivial normal subgroups .The question as to which infinite group s have this property P, say, was first raised by Philip Hall .It was shown in [2] that, if G is a finitely generated infinite P-group which contains a proper normal subgroup of finite index, then G is cyclic, and our conjecture is that Z is the only finitely generated infinite P -group which is no t simple .It was further remarked in [2] that is should perhaps be possible to deal with the locally soluble case, and this note represents a step in this direction .The following is proved .
Theorem .Let G be an infinite, locally soluble group which is isomorphic to all of its nontrivial normal subgroups .If G/G' has finite p -rank for all p = o or a prime then G is cyclic .
We recall that an abelian group A has p-rank r if the cardinality of a maximal independent subset of elements of A of order p is equal to r .In particular, if A has finite (Prüfer) rank then the p-ranks of A ar e boundedly finite .
There is one aspect of the proof of our theorem which recalls part of the proof from [2], namely the exploitation of "linearity conditions" which are forced by the rank restrictions (in conjunction with propert y P) .In the case where G has normal abelian p-sections of possibly infinit e rank, such a technique is bound to fail, and it is not clear how one might approach the case where, for example, G is an arbitrary locally nilpotent group with P .Clearly such a group is either torsionfree or a p-group , but beyond that there is little that we can say at the moment .

Proof of t he theorem :
Suppose first that G has a nontrivial, torsionfree soluble image S and let r be the O-rank of G /G' .Then, because of property P, S has finite Hirsch length (that is, the sum of the D--ranks of the derived factor s of G is finite) .Let F denote the Fitting subgroup of S .Then F is locally nilpotent and its abelian subgroups have finite D-rank .Since F is torsionfree, it is nilpotent (see Lemma 6 .37 of [3]) .Let A be a maximal normal abelian subgroup of F .Then A is self-centralizing in F and of rank at most r (again by P), and so F/A embeds in the group of (upper ) unitriangular r x r matrices over Q .It foliows that F j A and hence F has bounded rank and bounded nilpotency class c, say.Far each i = 1, . . ., c, let Zi denote the i -th term of the upper central series of F and let Di b e the centralizer in S of Zi jZi _ l (where Zo = = 1) .Then S /Di is a soluble group of automorphisms of Zi jZi _ ~, which is torsionfree abelian of rank at most r, and so S /Di embeds in GL(r, Q) .By the result of Zassenhau s ([3, Theorem 3 .23D,S /D i has bounded derived length .Let D = = flD .Then S /D has bounded derived length .Further, D stabilizes a series of length c in F and so, writing C for the centralizer of F in S, we see that D/C is nilpotent ([1, Lemma 3 .51) .But C = Z(F) (e .g .Lemma 2 .17 of [31) and so [C, D] = 1 and D is nilpotent and hence in F .It follows that S has bounded derived length and we can choose N minimal subject to N q G and G/N torsionfree soluble .If N 1 then, by property P, N has a nontrivial, torsionfree soluble image, contradicting the definitio n of N .Thus N = 1 and G is soluble .Clearly G Z in this case .From now on, we may assume that all soluble images of G are periodic . (I f G were to have a nonperiodic soluble image then some abelian normal factor of G would be nontrivial and torsionfree and so, again by P, G/G' would have a nontrivial torsionfree image .)Let H/K be an arbitrar y chief factor of G --such exists in every nontrivial group .Then H/ K is an elementary abelian pgroup, for some prime p, and we see that G therefore has a nontrivial finite p-image .Let P~= G'GP and, for i > 1, let Pi +1 = Pi' P p .By property P, the subgroups Pi form a strictly descending chain of normal subgroups of G .Also, each G /Pi is a finit e p-group .Let R = flPi and write _G = Gil?, _ Pi = Pi /R, i = 1, 2, . . . .

i=1 _
Let s be the rank of G /PI and let A be an arbitrary finitely generated abelian subgroup of G .The subgroups A n Pi form a descending chain , with trivial intersection, such that each A/A n P i is a finite (abelian) p-group of rank at most s (since A Pi jPi is subnormal in G jPi ) .It follows that A has rank at most s, and so G is a locally soluble grou p whose abelian subgroups have bounded rank .By a result of Merzljakov (see p .89, vol . 2 of [3] for a reference), G has finite rank .Now by Lemma 10 .39 of [3], G is periodic-by-soluble and hence periodic .Clearly, therefore, G is a locally nilpotent p-group and hence a Cernikov group (Corollary 1 to Theorem 6 .36 of [3]) .Since is residually finite, it must be finite, contradicting the choice of the subgroups P .This completes the proof of the theorem .■