Groups with many rewritable products

Author:

Curzio Mario,Longobardi Patrizia,Maj Mercede,Rhemtulla Akbar

Abstract

For any integer n 2 n \geq 2 , denote by R n {R_n} the class of groups G G in which every infinite subset X X contains n n elements x 1 , , x n {x_1}, \ldots ,{x_n} such that the product x 1 x n = x σ ( 1 ) x σ ( n ) {x_1} \ldots {x_n} = {x_{\sigma (1)}} \cdots {x_{\sigma (n)}} for some permutation σ 1 \sigma \ne 1 . The case n = 2 n = 2 was studied by B. H. Neumann who proved that R 2 {R_2} is precisely the class of centre-by-finite groups. Here we show that G R n G \in {R_n} for some n n if and only if G G contains an FC-subgroup F F of finite index such that the exponent of F / Z ( F ) F/Z(F) is finite, where Z ( F ) Z(F) denotes the centre of F F .

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference9 articles.

1. Rewriting products of group elements. I;Blyth, Russell D.;J. Algebra,1988

2. Rewriting products of group elements. II;Blyth, Russell D.;J. Algebra,1988

3. Rewritable products in FC-by-finite groups;Blyth, Russell D.;Canad. J. Math.,1989

4. On a combinatorial problem in group theory;Curzio, Mario;Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8),1983

5. A permutational property of groups;Curzio, Mario;Arch. Math. (Basel),1985

Cited by 2 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Groups with Many Rewritable Products;Communications in Algebra;2013-05-21

2. A permutability problem in infinite groups and Ramsey's theorem;Bulletin of the Australian Mathematical Society;2001-08

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