B.H. Neumann's question on ensuring commutativity of finite groups

Abstract

This paper is an attempt to provide a partial answer to the following question put forward by Bernhard H. Neumann in 2000: "Let G be a finite group of order g and assume that however a set M of m elements and a set N of n elements of the group is chosen, at least one element of M commutes with at least one element of N. What relations between g, m, n guarantee that G is Abelian?" We find an exponential function f(m, n) such that every such group G is Abelian whenever |G| > f(m, n) and this function can be taken to be polynomial if G is not soluble. We give an upper bound in terms of m and n for the solubility length of G, if G is soluble. Copyright Clearance Centre, Inc.