A COMBINATORIAL PROPERTY OF BURNSIDE VARIETY OF GROUPS OF FINITE EXPONENT

Abstract

Let w be a word in the free group on the set {x1, …, xn} and let [Formula: see text] be the variety of groups defined by the law w = 1. Define [Formula: see text], to be the class of all groups G in which for all infinite subsets Y1, …, Yn, there exist yi ∈ Yi such that w(y1, …, yn) = 1 and define [Formula: see text] (respectively [Formula: see text]) to be the class of all groups G in which for all infinite subset Y (respectively for all m-element subset Y) there exist n distinct elements y1, …, yn ∈ Y, such that w(y1, …, yn) = 1. In this paper we prove that [Formula: see text] and [Formula: see text] for some positive integer m, where w is a certain word and [Formula: see text] is the class of finite groups.