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Permutable word products in groups

Published online by Cambridge University Press:  17 April 2009

P.S. Kim  and
A.H. Rhemtulla
P.S. Kim
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
A.H. Rhemtulla
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
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Abstract

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Let u(x1,…,xn) = x11 … x1m be a word in the alphabet x1, …,xn such that x1i ≠ x1i for all i = 1,…, m − 1. If (H1, …, Hn) is an n-tuple of subgroups of a group G then denote by u(H1, …, Hn) the set {u(h1,…,hn) | hi ∈ Hi}. If σ ∈ Sn then denote by uσ(H1,…,Hn) the set u(Hσ(1),…,Hσ(n)). We study groups G with the property that for each n-tuple (H1,…,Hn) of subgroups of G, there is some σ ∈ Sn σ ≠ 1 such that u(H1,…,Hn) = uσ(H1,…,Hn). If G is a finitely generated soluble group then G has this property for some word u if and only if G is nilpotent-by-finite. In the paper we also look at some specific words u and study the properties of the associated groups.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 40 , Issue 2 , October 1989 , pp. 243 - 254
Copyright
Copyright © Australian Mathematical Society 1989

References

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