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 2 G such that w(ag; b) = 1, under what conditions does the group G belong to W ? We introduce for every g 2 G the sets Ww L (g) = fa 2 G j w(g; a) = 1g and Ww R (g) = fa 2 G j w(a; g) = 1g ; 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 2 G there exists g 2 G for which [ag; b] = 1. Hence when G is a Y -group and w is the commutator word [x; y], the set Ww L (g) = Ww R (g) is the centralizer of g in G, and the answer to the problem is a rmative. If G belongs to the class Y , we show that, more generally, the problem has a positive answer whenever each subset Ww L (g) is a subgroup of G, or equivalently, if each subset Ww R (g) is a subgroup of G. The sets Ww L (g) and Ww R (g) can be called the centralizer-like subsets associated with the word w. They need not be subgroups in general: we examine some su cient conditions on the group G ensuring that the sets Ww L (g) and Ww 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 Ww L (g) is a subgroup of G for every g 2 G and the class of all groups G for which each subset Ww R (g) is a subgroup... [edited by author]