A CONDITION ON FINITELY GENERATED

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1 COMMUNICATIONS IN ALGEBRA, 27(11), (1999) A CONDITION ON FINITELY GENERATED SOLUBLE GROUPS Alireza Abdollahi Department of Mathematics, University of Isfahan, Isfahan, Iran Bijan Taeri Department of Mathematics, University of Technology of Isfahan, Isfahan, Iran In this note we show that if G is a finitely generated soluble group, then every infinite subset of G contains two elements generating a nilpotent group of class at most k if and only if G is finite by a group in which every two generator subgroup is nilpotent of class at most k. 1. INTRODUCTION D Let X be a class of groups. We say that a groilp G satisfies the condition (X, (0) if every infinite set of elements of G contains a pair which generate a group in the class X. We use the notation A, N, and N k 5633 Copyright 1999 by Marcel Dekker, Inc.

2 5634 ABDOLLAHI AND TAERI - """ for the classes of abelian groups, nilpotent groups, and nilpotent groups of class at most k, respectively. For convenience, we say "of class k" where we mean "of class at most k". In response to a question of Paul Erdos, B. H. Neumann proved in [4] that the ~ondition (A, (0) is equivalent to being center-by-finite. This result has initiated a great deal of research. For example J. C. Lennox and J. Wiegold showed in [3] that a finitely generated soluble group satisfies the condition (N, (0) if and only if it is finite-bynilpotent. For a group G we denote by Zn(G) and fn(g), respectively, the (n + 1) -th term of the upper central series and the n -th term of the lower central series of G. C. Delizia in [1] and [2] studied the groups with the condition (N2, (0) and proved that a finitely generated soluble (or residually finite) group G satisfies the property (N 2,00) if and only if G/Z2(G) is finite. A question arises naturally: If G is a finitely generated soluble group with the property (Nk, (0), for k::::: 3, is IG/Zk(G)1 < oo? This is not true in general, but we can prove it for metabelian groups (see Proposition 1). It is well known that if IG/Zk(G)1 < 00 then lfk+l(g)1 < 00, for any group G (see of [7]). Since G/fk+1(G) is a nilpotent group of class k, every two generator subgroup is of class k. We denote by NP) the class of all groups in which every two generator subgroup is nilpotent of class k. Thus if IG/ Zk( G) I < 00 then G is a finite-by-np) group. Of course the converse is false (see the example in Proposition 1). Considering this weaker condition we are able to prove Theorem. Let G be a finitely generated soluble group. Then G has the property (Nk, (0) if and only if G is a finite-by-np) group. 2. PROOFS Proposition 1. Let G be a finitely generated metabelian group. Then G has the property (Nk, (0) if and only if G/ Zk( G) is finite. This result is false for finitely generated soluble groups with derived length ::::: 3. Proof. Suppose G has the property (Nk, (0). We assume G to be torsion free nilpotent and first prove that G = Zk(G). Thus, without loss of

3 _ _~--~-~ CONDITION ON FINITELY GENERATED SOLUBLE GROUPS , generality we may assume that G E N k +1' Let 1 =f. x, Y E G, then the set is infinite, and there exist i,j with i < j such that (Xiy,xjy) E Nk. For h=j-i,wehave Xh=xjy(xiy)-I so (xh,xiy) ENk and [XiY'kXh]=1. Thus But G is torsion free so [y, kx] = 1. TherefQre G is a k-engel group. Now since G is metabelian, by a well known Theorem fk+i(g) has finite exponent (see Theorem 7.36 of [6]). But G is torsion free, so fk+1(g) = 1, and thus G = Zk(G), as required. Now we assume G is nilpotent. Let T be the torsion subgroup of G, thus T is a finitely generated torsion nilpotent group, hence it is finite. Now GjT is torsion free, so GjT = Zk(GjT) that is fk+1(gjt) = 1 and fk+i(g) :s:; T is finite. Since G is finitely generated, GjZk(G) is finite. Now consider the general case. We know from Theorem A of [3] that G is finite-by-nilpotent. So there exists a finite normal subgroup H of G such that Gj H is nilpotent. Therefore f k+1(g j H) = f k+1 (G)H j H is finite. Since H is finite, so is fk+1(g). Thus GjZk(G) is finite. Conversely, if Gj Zk( G) is finite and {Xi lie I} is an infinite set of elements of G, then there exist i,j E I, i =f. j, Xi == Xj mod Zk(G). So z=xixtezk(g),thus (Xi,Xj) = (z,xj) is nilpotent of class k and G satisfies the property (Nk, (0). '--:-.:::: ====~. Now if the derived length 2: 3 the above result is false: M. F. Newman in [5] has constructed a group G, which is the splitting extension of A by B, where A is a free abelian group of rank 2m-I, and B is a certain subgroup of Aut A, which is free nilpotent of class 2 of rank m - 1. G is a finitely generated torsion free nilpotent group of class exactly m in which every m - 2r generator subgroup is nilpotent of class m - r, for each non-negative integer r (less than mj2). Note that G has derived length 3. If m is an even integer 2: 4 we can choose r = 'i Thus every two generator subgroup of G has class k = m - r = 'i- + 1 < m, and G has class exactly m. So G/ Zk( G) can not be finite. Note that if m = 4,6,8,... we have k = 3,4,5,... Thus for every k 2: 3 we have D

4 5636 ABDOLLAHI AND TAERI a finitely generated torsion free nilpotent group G with derived length 3 having the property (Nk, 00), such that GIZk(G) is infinite. 0 To prove the Theorem we need the following key lemma Lemma 2. Let G = (x, y) be a torsion free nilpotent (Nk, 00) group. Then G = Zk(G). Proof. By induction on the nilpotency class of G we may assume that G = Zk+1(G). We show that every commutator of weight k + 1 in {x, y} is trivial. We may assume 1 #- x and 1 #- y. Since G is torsion free the set is infinite. So there exist i,s with i < s such that (xiy,xsy) is nilpotent of class k. For r = s - i, we have x T = x S Y(X i y)-l so that (xt,xsy) is nilpotent of class k. Since (x, xsy) is torsion free nilpotent, it follows from (xt,xsy)enk that (x,xsy)enk. But (x,xsy)=(x,y)=g and hence GENk D Proof of the Theorem. By Theorem A of [3] there exists a finite normal subgroup H of G such that GI H is nilpotent. Let T I H be the torsion subgroup of GIH. Then GIT is a torsion free nilpotent (Nk,oo) group. Therefore, by Lemma 2, GIT is NP) -group. Also T is finite, as T I H is a finitely generated soluble torsion group and H is finite. Thus G is a finite-by-np) group. Conversely, suppose H is a finite normal subgroup of G such that G IH is N~2)_group. By Theorem A of [3] GIH is finite-by-nilpotent. Since H is finite, G is finite-by-nilpotent. Thus G is residually finite. Therefore there is a normal subgroup N of G with finite index such that H n N = 1. Let X be an infinite subset of G, then X n gn is an infinite set for some 9 E G and there exist two distinct elements x, y E X n gn such that (x'yh E N k. Also (x,~n is cyclic. Therefore and the result follows. 0

5 .. CONDITION ON FINITELY GENERATED SOLUBLE GROUPS 5637 Let n be a positive integer and X be a class of groups. We say that a group satisfies the condition (X, n) if any set of n + 1 elements of the group contains a pair which generate a group in class X. It follows from Theorem A of [3] that, if G is a finitely generated soluble group with the condition (N, 00), then G satisfies the condition (N, n), for some n. In the proof of the Theorem, considering n = IGINI, we have Corollary 3. Let G be a finitely generated infinite soluble group satisfying the property (N k, 00), k > 0, then G has the property (Nk, n), for some n. ACKNOWLEDGEMENTS The authors would like to thank the referee for some valuable suggestions. Also they whish to thank their supervisor Dr. A. Mohammadi Hassanabadi for his help and encouragement while doing this work. REFERENCES [1] [2] [3] ~ [4] [5] C. Delizia, Finitely Generated Soluble Groups with a Condition on Infinite Subsets, Instito Lombardo (Rend. Sc. ) A 128 (1994) C. Delizia, On Certain Residually Finite Groups, Comm. Algebra, 24 (1996) J. C. Lennox and J. Wiegold, Extension of a Problem of Paul Erdos on Groups, J. Austral. Math. Soc., 31 (1981) B. H. Neumann, A Problem of Paul Erdos on Groups, J. Austral. Math. Soc. (Serise A), 21 (1976) M. F. Newman, Some Varieties of Groups, J. Austral. Math. Soc., Vol XVI, Part 4 (1973) D [6] D. J. S. Robinson, Finiteness C.onditions and Generalized Soluble Groups, Part II, Springer - Verlag, Berlin (1972).

6 D ~_.-, ABDOLLAHI AND TAERI [7] D. J. S. Robinson, A Course in the Theory of Groups, Springer - Verlag, Berlin (1982). Received: March 1998 Revised: October 1998

MINIMAL NUMBER OF GENERATORS AND MINIMUM ORDER OF A NON-ABELIAN GROUP WHOSE ELEMENTS COMMUTE WITH THEIR ENDOMORPHIC IMAGES

MINIMAL NUMBER OF GENERATORS AND MINIMUM ORDER OF A NON-ABELIAN GROUP WHOSE ELEMENTS COMMUTE WITH THEIR ENDOMORPHIC IMAGES Communications in Algebra, 36: 1976 1987, 2008 Copyright Taylor & Francis roup, LLC ISSN: 0092-7872 print/1532-4125 online DOI: 10.1080/00927870801941903 MINIMAL NUMBER OF ENERATORS AND MINIMUM ORDER OF