A Property Equivalent to n-permutability for Infinite Groups

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1 Journal of Algebra 221, (1999) Article ID jabr , available online at on A Property Equivalent to n-permutability for Infinite Groups Alireza Abdollahi* and Aliakbar Mohammadi Hassanabadi Department of Mathematics, University of Isfahan, Isfahan, Iran and Bijan Taeri Department of Mathematics, University of Technology of Isfahan, Isfahan, Iran Communicated by Gernot Stroth Received September 7, 1998 Let n be an integer greater than 1. A group G is said to be n-permutable whenever for every n-tuple x 1 x n of elements of G there exists a non-identity permutation σ of 1 n such that x 1 x n = x σ 1 x σ n In this paper we prove that an infinite group G is n-permutable if and only if for every n infinite subsets X 1 X n of G there exists a non-identity permutation σ on 1 n such that X 1 X n X σ 1 X σ n 1999 Academic Press 1. INTRODUCTION Permutable groups have been studied by various people (for example, see ). Let n be an integer greater than 1. Recall that a group G is called n-permutable whenever for every n-tuple x 1 x n of elements of G there exists a non-identity permutation σ of 1 n such that x 1 x n = x σ 1 x σ n. Also a group is said to be permutable if it is n-permutable for some integer n>1. The main result for groups in this class was obtained by Curzio et al. in [3], where it was shown that such * abdolahi@math.ui.ac.ir. aamohaha@math.ui.ac.ir. b.taeri@cc.iut.ac.ir /99 $30.00 Copyright 1999 by Academic Press All rights of reproduction in any form reserved. 570

2 n-permutability 571 groups are finite-by-abelian-by-finite. Let n > 1 and m be positive integers. Let S n denote the group of all permutations on the set 1 n. A natural extension of permutable groups, namely m n -permutable groups, groups in which X 1 X n σ S n \1 X σ 1 X σ n for all subsets X i of G where X i =m for all i = 1 n, was introduced by Mohammadi Hassanabadi and Rhemtulla in [9]. It was proved there that such a group either is n-permutable or is finite of order bounded by a function of m and n. In [8] Mohammadi Hassanabadi investigated another extension of m n -permutable groups as follows. For positive integers n>1 and m a group G is called restricted m n -permutable if X 1 X n σ S n \1 X σ 1 X σ n for all subsets X i of G where X i =m for all i = 1 n. It was proved there that such a group is finite-by-abelian-by-finite. In [4] Longobardi et al. called a group G a Pn -group (n an integer greater than 1) if for every sequence X 1 X n of infinite subsets of G there exist x i in X i such that x 1 x n = x σ 1 x σ n for some non-trivial permutation σ in S n. They proved that every infinite Pn -group is an n-permutable group. Here we deal with another extension of infinite restricted m n -permutable and Pn -groups. Let n be an integer greater than 1. We call a group G a restricted n permutable group if X 1 X n σ S n \1 X σ 1 X σ n for all infinite subsets X 1 X n of G. Our main result is the following, which sharpens and generalizes that of [8] and also generalizes the result of [4] concerning Pn. Every infinite restricted n -permutable group is n-per- Theorem. mutable. 2. PROOFS To prove the theorem, we need the following results. Lemma 2.1. Let G be an infinite residually finite group which is a restricted n -permutable group. Then G is an n-permutable group. Proof. Let x 1 x n be arbitrary elements of G and S = { x 1 x n x σ 1 x σ n 1 σ S n \1 } Suppose, for a contradiction, that 1 S. Since G is residually finite and S is finite, there exists a normal subgroup N of finite index in G such that S N =. Now considering infinite subsets Nx 1 Nx n, there exists σ S n \1 such that Nx 1 Nx n Nx σ 1 Nx σ n and so x 1 x n x σ 1 x σ n 1 N, which is a contradiction.

3 572 abdollahi, mohammadi hassanabadi, and taeri Lemma 2.2. Let G = Dr i I G i be an infinite direct product of non-abelian subgroups. Then G is not a restricted n -permutable group for all integers n>1. Proof. Suppose, for a contradiction, that G is a restricted n permutable group for some integer n > 1. We show that G is an n- permutable group, which contradicts Corollary 2.9 of [1]. Let x 1 x n G and put S = x 1 x n x σ 1 x σ n 1 σ S n \1. Letk be any integer greater than S. Since G is an infinite direct product of normal subgroups, there exist k infinite normal subgroups N 1 N k of G such that N i N j = 1 for all distinct i j 1 k. Let l 1 k and consider infinite subsets N l x 1 N l x n. By the hypothesis, there exists σ l S n \1 such that x 1 x n x σl 1 x σl n 1 N l Therefore there exist two distinct i j 1 k and an element s S such that s N i N j = 1 and so G is an n-permutable group. We denote by A 1 the set a 1 a A for any non-empty subset A of a group. Let a and g be arbitrary elements of a group G. We define S a g = x G a x = g which is either an empty set or a right coset of the centralizer of a in G. A key result required in the proof of the theorem is the following: Lemma 2.3. Let G be an infinite restricted n -permutable group. Then the FC-centre of G is non-trivial. Proof. Suppose, for a contradiction, that the FC-centre of G is trivial. We construct n infinite subsets X 1 X n of G such that X 1 X n X σ 1 X σ n = for all non-identity permutations σ in S n. For this, for each m we construct n subsets X i m = a i 1 a i m of G (i = 1 n), such that X 1 m X n m X σ 1 m X σ n m = # for all non-identity permutations σ in S n. We argue by induction on m. Let m = 1. By Lemma 2.1 in [3], G is not an n-permutable group and so there exist a 1 1 a n 1 G such that a 1 1 a n 1 a σ 1 1 a σ n 1 for all σ S n \1. Now suppose that we have already defined subsets X i m = a i 1 a i m of G (i = 1 n) satisfying # for all σ S n \1. Suppose that we have already defined a i m+1 and so X i m+1 for i = 1 r such that for all σ S n \1 X 1 m+1 X 2 m+1 X r m+1 X r+1 m X n m X σ 1 j1 X σ n jn =

4 n-permutability 573 where j t = m + 1 whenever σ t 1 r and otherwise j t = m. Let T r+1 be the union of all the following sets where σ varies over S n \1, X 1 σ i 1 s i 1 X 1 σ 1 s 1 X 1 m+1 X r m+1 X r+1 m X n m X 1 σ n s n X 1 σ i+1 s i+1 where s l = m + 1 whenever σ l 1 r and otherwise s l = m; also i varies over 1 n and if i = 1ori = n then we define respectively X 1 σ i 1 s i 1 X 1 σ 1 s 1 = 1 or X 1 σ n s n X 1 σ i+1 s i+1 = 1. Also put ( U r+1 = X 1 r m+1 X 1 1 m+1 T r+1 1 σ S n X σ 1 j1 X σ n jn ) X 1 n m X 1 r+2 m where j l = m + 1 whenever σ l 1 r and otherwise j l = m. Now we prove that there exists an element a r+1 m+1 G\U r+1 such that if X r+1 m+1 = a r+1 1 a r+1 m+1 then for all σ S n \1 X 1 m+1 X r+1 m+1 X r+2 m X n m X σ 1 j1 X σ n jn = where j l = m + 1 whenever σ l 1 r + 1 and otherwise j l = m. Suppose not. Therefore a 1 i1 a 2 i2 a n in = a σ 1 j1 a σ 2 j2 a σ n jn for some 1 i 1 i r+1 m i r+2 m 1 j 1 m, and 1 j s m + 1 whenever σ s 1 r + 1. Suppose that σ t =r + 1. If i r+1 m + 1orj t m + 1 then we get contradiction with the induction hypothesis or the choice of a r+1 m+1. Therefore we must always have i r+1 = j t = m + 1 and so ( ) 1 a r+2 ir+2 a n in aσ t+1 jt+1 a σ n jn = a 1 r+1 i r+1 (a 1 i1 a r ir ) 1 a σ 1 j1 a σ t 1 jt 1 a r+1 jt Now we define g σ and f σ for all σ S n \1as ( ) 1 a r+2 ir+2 a n in aσ t+1 jt+1 a σ n jn if 1 t n 1 f σ = a r+2 ir+2 a n in if t = n and ( ) 1aσ 1 j1 a1 i1 a r ir a σ t 1 jt 1 if 2 t n g σ = ( ) 1 a1 i1 a r ir if t = 1 where t = σ 1 r + 1. Hence a r+1 m+1 S g σ f σ and so ( ) G = U r+1 S gσ f σ

5 574 abdollahi, mohammadi hassanabadi, and taeri where σ in varies over the set of all non-identity permutations in S n such that S g σ f σ. Obviously the set of pairs g σ f σ is finite. Therefore shows that G is a finite union of right cosets of the centralizers of g σ s. Thus by the famous theorem of Neumann [10] there exists g σ in the FC-centre of G such that S g σ f σ. But by the hypothesis g σ = f σ = 1. Thus there exist n 1 -tuples i 1 i r i r+2 and j 1 j t 1 j t+1 where 1 i 1 i r m + 1, 1 i r+2 m, t = σ 1 r + 1, and j l = m + 1 whenever 1 σ l r and otherwise j l = m such that ( ) 1 a r+2 ir+2 a n in a σ t+1 jt+1 a σ n jn = ( a 1 i1 a r ir ) 1aσ 1 j1 a σ t 1 jt 1 = 1 So for any a X r+1 m we have the following, which contradicts the induction hypothesis: a 1 i1 a r ir aa r+2 ir+2 a n in = a σ 1 j1 a σ 2 j2 a σ t 1 jt 1 aa σ t+1 jt+1 a σ n jn Therefore we have defined X r+1 m+1. Thus we have inductively defined X i m = a i 1 a i m for all m such that for all σ S n \1 X 1 m X n m X σ 1 m X σ n m = Now set X i = m=1 X i m (i = 1 n), then X i is infinite and X 1 X n X σ 1 X σ n = for all σ S n \1. Otherwise there exist n-tuples i 1 and j 1 on and π S n \1 such that a 1 i1 a n in = a π 1 j1 a π n jn Let s = Max i 1 j 1. Then X 1 s X n s X π 1 s X π n s, which is a contradiction with the construction of X i s (i = 1 n). By Lemma 2.3, every non-trivial restricted n -permutable group has a non-trivial FC-element and since the class of restricted n permutable groups is closed under homomorphic images we have: Every restricted n -permutable group is FC-hyper- Corollary 2.4. central. Lemma 2.5. Let G be an infinite restricted n -permutable group. If G is finitely generated or non-periodic then G is an n-permutable group.

6 n-permutability 575 Proof. Suppose that G is finitely generated. By Corollary 2.4, G is FChypercentral. Now by a result of McLain [7] (or see p. 133 of [11]) a finitely generated FC-hypercentral group is nilpotent-by-finite. Therefore G is a finitely generated nilpotent-by-finite group and so G is residually finite. Thus G is n-permutable by Lemma 2.1. Now assume that G is non-periodic. Then there is an element x of infinite order in G.Letx 1 x n be arbitrary elements of G. By the previous part x x 1 x n is an n-permutable group and so G is n-permutable. Lemma 2.6. Let G be a restricted n -permutable group. Then G is hyperabelian-by-finite. Proof. We may assume that G is infinite, and it suffices to show that G contains a non-trivial normal abelian subgroup. Suppose no such normal abelian subgroup exists, and let x be a non-identity element in the FC-centre of G which exists by Lemma 2.3. Let N 1 = x G be the normal closure of x in G, and let C = C G N 1. Then G C is finite and N 1 C = Z N 1 is a normal abelian subgroup of G. Hence N 1 C = 1. Therefore N 1 is finite and, having a trivial centre, it is certainly non-abelian. Now suppose, inductively, that we have already defined normal non-abelian finite subgroups N 1 N t of G such that N 1 N t generate their direct product in G. Write D = C G N 1 N t ; thus G D is finite. Now using Lemma 2.3 we can choose a non-trivial element y in the FC-centre of D. Then y is an element of the FC-centre of G. LetN t+1 = y G. It is easily seen that N t+1 is a finite non-abelian group. Moreover, N t+1 D, so that N 1 N t N t+1 generate their direct product in G. Thus we have found in G an infinite direct product N 1 N 2 N t of finite non-abelian groups, which together with Lemma 2.2 gives a contradiction. Lemma 2.7. Let G be an infinite restricted n -permutable group which is not Černikov. Then G is an n-permutable group. Proof. By Lemma 2.5, we may assume that G is periodic. By Lemma 2.6, there exists a normal hyperabelian subgroup H of finite index in G. Therefore H is a periodic locally soluble group and G is locally finite. Let x 1 x n be arbitrary elements in G and let A be the finite subgroup generated by x 1 x n. We note that H is not a Černikov group and A can be regarded as a finite group of automorphisms of H. Now by a result of Zaicev [13], there exists an abelian subgroup B of H which is not Černikov and B is a normal subgroup of AB. Since B is periodic it is a direct product of the Sylow p-subgroups B p of B. If infinitely many B p are non-trivial, then since A has only finitely many prime divisors, there exists an infinite subgroup D of B which is normal in AB such that A D = 1. Consider the n infinite subsets Dx 1 Dx n. By the hypothesis there exists

7 576 abdollahi, mohammadi hassanabadi, and taeri σ S n \1 such that Dx 1 Dx n Dx σ 1 Dx σ n and so x 1 x n x σ 1 x σ n 1 A D = 1. Therefore x 1 x n = x σ 1 x σ n as required. So assume that there exist only finitely many B p which are non-trivial. Since B is not a Černikov group and since the product of two normal Černikov subgroups of a group is a Černikov group, then there exists a prime number p such that B p is not Černikov. Thus by Theorem of [12], C = b B b p = 1 is an infinite elementary abelian p-group. Clearly C is normal in AB. Now the infinite group AC is a residually finite-by-finite group and so AC is residually finite. Therefore by Lemma 2.1, AC is an n-permutable group and the proof is complete. We need the following remark in the final step of the proof of the theorem. Here x denotes the order of an element x of a group. Remark 2.8. We note that if x 1 x n (n>1) are p-elements (p a prime) of distinct orders in an abelian group then r< x 1 x n t where r = Min x 1 x n and t = Max x 1 x n. Proof of the Theorem. Let G be an infinite restricted n -permutable group. By Lemma 2.7, we may assume that G is a Černikov group. Thus there exists an infinite normal subgroup A of G which is a direct product of finitely many groups isomorphic to C p, the quasicyclic p-group, for some prime number p. Letx 1 x n G and let X be the finite subgroup generated by x 1 x n (we note that G is locally finite). Let Y be the group of automorphisms of A induced by the elements of X under conjugation. Then Y is finite. Let α 0 be an integer such that a p α 0 for any a X A. By Lemma 3.5 of [4] there are infinite sequences α 0 <α 1 < of integers and a 1 a 2 of elements of A such that for any i, a i =p α i, and ai y >p α i 1, for any y Y \CY a i. Now partition the set a i i 1 into n infinite disjoint subsets J i, i = 1 n. Consider the set J i x i, i = 1 n, and let σ S n \1 be such that a i1 x 1 a in x n = a j1 x σ 1 a jn x σ n for suitable a i1 J 1 a in J n and a j1 J σ 1 a jn J σ n. Therefore x = x 1 x n 1 x σ 1 x σ n = a x 1 x n i 1 a x ( x n σ 1 x σ n a j 1 a x ) σ n 1 We note that i 1 are pairwise distinct as are j 1.If i 1 j 1 =

8 n-permutability 577 then by Remark 2.8, p α r < x where r = Min i 1 j 1, which is a contradiction, since α r >α 0 and x X A. Thus F = i 1 j 1.Let F =s. We may assume, without loss of generality, that i 1 = j 1 i s = j s. Then we may write x = a i1 y 1 z1 a is y s z s a y s+1 i s+1 a y ( n z a s+1 j s+1 a z ) n 1 for some y 1 y n z 1 z n X. Now suppose, for a contradiction, that x 1. If a i1 y 1 = = a is y s =1 then s<n, since x 1. Then since i s+1 j s+1 are pairwise distinct, by Remark 2.8, x >p α k where k = Min i s+1 j s+1, which is a contradiction. Thus we may assume, without loss of generality, that y l Y \C Y a il, for l = 1 s and i 1 < <i s. Now we claim that the elements [ ] [ ] y ai1 y 1 ais y s a s+1 i s+1 a y n a z s+1 j s+1 a z n have distinct orders. For, since p α i l 1 < a il y l p α i l for l = 1 s and α i1 < <α is, then the elements a i1 y 1 a is y s have distinct orders. Clearly a y s+1 i s+1 a y n a z s+1 j s+1 a z n have distinct orders. If there exist l 1 s and k s + 1 n such that a il y l = a ik or a il y l = a jk then since p α i l 1 < a il y l p α i l, αil = α ik or α il = α jk and so i l = i k or i l = j k, a contradiction. Now by Remark 2.8, p t < x, where p t = Min a i1 y 1 a is y s a y s+1 i s+1 a y n a z s+1 j s+1 a z n But t>α 0 which is a contradiction. REFERENCES 1. R. D. Blyth, Rewriting products of group elements, I, J. Algebra 116 (1988), R. D. Blyth, Rewriting products of group elements, II, J. Algebra 119 (1988), M. Curzio, P. Longobardi, M. Maj, and D. J. S. Robinson, A permutational property of groups, Arch. Math. 44 (1985), P. Longobardi, M. Maj, and A. H. Rhemtulla, Infinite groups in a given variety and Ramsey s theorem, Comm. Algebra 20 (1992), P. Longobardi, M. Maj, and S. E. Stonehewer, The classification of groups in which every product of four elements can be reordered, Rend. Sem. Mat. Univ. Padova 93 (1995), M. Maj, On the derived length of groups with some permutational property, J. Algebra 136 (1991), D. H. McLain, Remarks on the upper central series of a group, Proc. Glasgow Math. Assoc. 3 (1956), A. Mohammadi Hassanabadi, A property equivalent to permutability for groups, Rend. Sem. Mat. Univ. Padova, 100 (1998), A. Mohammadi Hassanabadi and A. H. Rhemtulla, Criteria for commutativity in large groups, Canad. Math. Bull. 41, No. 1 (1998),

9 578 abdollahi, mohammadi hassanabadi, and taeri 10. B. H. Neumann, Groups covered by permutable subsets, J. London Math. Soc. 29 (1954), D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Part 1, Springer-Verlag, Berlin, D. J. S. Robinson, A Course in the Theory of Groups, 2nd ed., Springer-Verlag, Berlin, D. I. Zaicev, On solvable subgroups of locally solvable groups, Dokl. Akad. Nauk SSSR 214 (1974), [Translation in Soviet Math. Dokl. 15 (1974), ]