Locally soluble groups with all nontrivial normal subgroups isomorphic

Publicacions Matemàtiques, Vol 38 (1994), 203-205 . LOCALLY SOLUBLE GROUP S WITH ALL NONTRIVIA L NORMAL SUBGROUPS ISOMORPHI C JOHN C . LENNOX, HOWARD SMITH AND JAMES wIEGOL D A bstract  Let G be an infinite, locally soluble group which is isomorphic t o all its nontrivial normal subgroups . If G/G ' has finite p-rank fo r p = 0 and for all primes p, then G is cyclic . 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 show n 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 possibl e to deal with the locally soluble case, and this note represents a step i n 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 -ran k 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 o f a maximal independent subset of elements of A of order p is equal t o 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 par t 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 migh t approach the case where, for example, G is an arbitrary locally nilpotent 204  J . Ú . LENNOX, H . SMITH, J . WIEGOL D 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 h e 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 hasfinit e 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 i s locally nilpotent and its abelian subgroups have finite D-rank . Since F i s torsionfree, it is nilpotent (see Lemma 6 .37 of [3]) . Let A be a maxima l normal abelian subgroup of F . Then A is self-centralizingin F and o f 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 ha s bounded rank and bounded nilpotency class c, say . Far each i = 1, . . . , c , let Z i denote the i -th term of the upper central series of F and let D i b e the centralizer in S of Z i jZi _ l (where Zo = = 1) . Then S /D i is a solubl e group of automorphisms of Z i jZi _ ~ , which is torsionfree abelian of ran k at most r, and so S /D i 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 . i= 1 Then S /D has bounded derived length . Further, D stabilizes a series o f length c in F and so, writing C for the centralizer of F in S, we see tha t D/C is nilpotent ([1, Lemma 3 .51) . But C = Z(F) (e .g . Lemma 2 .17 o f [31) and so [C, D] = 1 and D is nilpotent and hence in F . It follows tha t S has bounded derived length and we can choose N minimal subject t o 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 norma l 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 tha t G therefore has a nontrivial finite pimage . Let P~ = G'G P and, fo r i > 1, let P i +1 = P i ' P p . By property P, the subgroups P i form a strictl y descending chain of normal subgroups of G . Also, each G /P i is a finit e p-group . Let R = flP i and write _G = Gil?, _ Pi = P i /R, i = 1, 2, . . . . i=1  _ Let s be the rank of G /P I and let A be an arbitrary finitely generate d 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 GROUPS WITH NORMAL SUBGROUPS ISOMORPHIC  20 5 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 Merzljako v (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 subgroup s P . This completes the proof of the theorem . ■ Reference s 1. PHILIP HALL, "The Edmonton notes on nilpotent groups, " Quee n Mary College Mathematics Notes, London, 1969 . 2. JOHN C . LENNOX, HOWARD SMITH AND JAMES WIEGOLD, A problem about normal subgroups, Journal of Pure and Applied Algebr a 88 (1993), 169-171 . 3. D . J . S . ROBINSON, "Finiteness conditions and generalized solubl e groups," Springer - Verlag, Berlin - Heidelberg-New York, 1972 . John C . Lennox :  Howard Smith : School of Mathematics  Department of Mathematic s University of Wales  Bucknell Universit y College of Cardiff  Lewisburg PA 1783 7 Cardiff CF2 4AG  U .S .A . WALES James Wiegold : School of Mathematic s University of Wale s College of Cardif f Cardiff CF2 4A G WALE S Rebut el 14 d'Octubre de 1993