## 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]