Some group properties associated with two-variable words

Abstract

Let w(x, y) be a word in two variables and W the variety determined by w. In this thesis, which includes a work made in collaboration with C. Nicotera [5], we raise the following question: if for every pair of elements a, b in a group G there exists g ∈ G such that w(a, b) = 1, under what conditions does the group G belong to W ? We introduce for every g ∈ G the sets W L (g) = {a ∈ G | w(g, a) = 1} and W R (g) = {a ∈ G | w(a, g) = 1} , where the letters L and R stand for left and right. In [2], M. Herzog, P. Longobardi and M. Maj observed that if a group G belongs to the class Y of all groups which cannot be covered by conjugates of any proper subgroup, then G is abelian if for every a, b ∈ G there exists g ∈ G for which [a, b] = 1. Hence when G is a Y -group and w is the commutator word [x, y], the set W L (g) = W w R (g) is the centralizer of g in G, and the answer to the problem is affirmative. If G belongs to the class Y , we show that, more generally, the problem has a positive answer whenever each subset W L (g) is a subgroup of G, or equivalently, if each subset W R (g) is a subgroup of G. The sets W w L (g) and W w R (g) can be called the centralizer-like subsets associated with the word w. They need not be subgroups in general: we examine some sufficient conditions on the group G ensuring that the sets W L (g) and W w R (g) are subgroups of G for all g in G. We denote by W w L and W w R respectively the class of all groups G for which the set W L (g) is a subgroup of G for every g ∈ G and the class of all groups G for which each subset W R (g) is a subgroup. 1