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On n-abelian groups and their generalizations

Published online by Cambridge University Press:  05 July 2011

By
Costantino Delizia  and
Antonio Tortora
Edited by
C. M. Campbell ,
M. R. Quick ,
E. F. Robertson ,
C. M. Roney-Dougal ,
G. C. Smith  and
G. Traustason
Costantino Delizia
Affiliation:
Università di Salerno Via Ponte don Melillo, Italy
Antonio Tortora
Affiliation:
Università di Salerno Via Ponte don Melillo, Italy
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
G. C. Smith
Affiliation:
University of Bath
G. Traustason
Affiliation:
University of Bath
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Summary

Abstract

For any integer n ≠ 0, 1, a group G is said to be n-abelian if it satisfies the identity (xy)n = xnyn. More generally, G is called an Alperin group if it is n-abelian for some n ≠ 0, 1. We consider two natural ways to generalize the concept of n-abelian group: the former leads to define n-soluble and n-nilpotent groups, the latter to define n-Levi and n-Bell groups. The main goal of this paper is to present classes of generalized n-abelian groups and to point out connections among them. Besides, Section 5 contains unpublished combinatorial characterizations for Bell groups and for Alperin groups. Finally, in Section 6 we mention results of arithmetic nature.

Keywords:n-abelian groups, Alperin groups, Bell groups, infinite subsets.

Introduction

Let n ≠ 0, 1 be an integer and let G be a group. In [4], R. Baer introduced the n-centre Z(G, n) of G as the set of all elements x ∈ G which n-commute with every element in the group, i.e. (xy)n = xnyn and (yx)n = ynxn for any y ∈ G. Later, in [20], L.-C. Kappe and M. L. Newell proved that (xy)n = xnyn for any y ∈ G if and only if (yx)n = ynxn for any y ∈ G. Thus one only n-commutative condition suffices to define Z(G, n). The n-centre is a characteristic subgroup and shares many properties with the centre (see, for instance, [4] and [20]).

Type
Chapter
Information
Groups St Andrews 2009 in Bath , pp. 244 - 255
Publisher: Cambridge University Press
Print publication year: 2011

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