Our concern in this paper is to obtain information about the structure of centralizers of elements of locally finite simple groups, in the light of the classification of finite simple groups (CFSG). This classification, in the form that the number of sporadic simple groups is finite, is frequently used, as seems

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Abstract. Let w be a word in the free group of rank n ∈ N and let V(w) be the variety of groups defined by the law w = 1. Define V(w ∗ ) to be the class of all groups G in which for any infinite subsets X1,..., Xn there exist xi ∈ Xi, 1 ≤ i ≤ n, such that w(x1,..., xn) = 1. Clearly, V(w) ∪ F ⊆ V(w ∗); F being the class of finite groups. In this paper, we investigate some words w and some certain classes P of groups for which the equality (V(w) ∪ F) ∩ P = P ∩ V(w ∗ ) holds. Introduction and results Let w be a word in the free group of rank n ∈ N and let V(w) be the variety of groups defined by the law w = w(x1,...,xn) = 1. P. Longobardi, M. Maj and A. Rhemtulla in [29] defined V(w ∗ ) to be the class of all groups G in which for any infinite subsets X1,..., Xn there exist xi ∈ Xi, 1 ≤ i ≤ n, such that w(x1,...,xn) = 1 and raised the question of whether V(w) ∪ F = V(w ∗ ) is true; F being the class