A Combinatorial Problem in Infinite Groups

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 of finite groups. There is no example, so far, of an infinite group in V(w)\V(w). In fact the origin of this problem is the following observation: Let G be an infinite group such that in every two infinite subsets ofG there exist two commuting elements, then G is abelian. This is an immediate consequence of the answer of B. H. Neumman to a question of P. Erdös; B. H. Neumman proved that an infinite group G is centre-by-finite if and only if every infinite subset of G contains two distinct commuting elements [37]. Since this first paper, problems of a similar nature have been the object of several articles (for example [2], [3], [5], [9], [11], [12], [15], [24], [27], [28], [39]). As far as we know, the equality V(w) ∪ F = V(w) is known for the following words: w = x, w = [x1, . . . , xn] [29], w = [x, y] 2 [26], w = [x, y, y] [41], w = [x, y, y, y] [42], w = (xy)xy [1], w = x1 1 · · ·x αm m where α1, . . . , αm are non-zero integers [4], w = (xy) (yx) or w = [x, y] where m ∈ {3, 6} ∪ {2 | k ∈ N} [6], w = [x, y][x, y] where n ∈ {±2, 3} [43] and w = [x, y] or w = (x1 x m 2 · · ·x m n ) 2 where m ∈ {2 | k ∈ N} [8]. In [38], P. Puglisi and L. S. Spiezia proved that every infinite locally finite group (or locally soluble group) in V([x,k y] ) is a k-Engel group; (recall that [x,k y] is defined inductively by [x,0 y] = x and [x,k y] = [[x,k−1 y], y] for k ∈ N). In [10], C. Delizia proved the equality V(w)∪F = V(w ) on the classes of hyperabelian, locally soluble and locally finite groups where w = [x1, . . . , xk, x1] and k is an integer greater than 2. Later G. Endimioni generalized these results by proving that every infinite locally finite or locally soluble group in V(w) belongs to the variety V(w), where w is a word in a free group such that finitely generated soluble groups in V(w) are nilpotent (see Theorem 3 of [14]) (recall that the variety V([x1, . . . , xk, x1]) (k > 2) is exactly the variety of nilpotent groups of nilpotency class at most k [35] and every finitely generated soluble Engel group is nilpotent [17].) We say that a group G is locally graded if and only if every finitely generated non-trivial subgroup of G has a non-trivial finite quotient. We proved in Theorem 4 of [3] that an infinite locally graded group in V([x1,k x2] ) is a k-Engel group. We generalize this result as Theorem A, below. In order to state our first result we need the following definition. Following [20] we say that a group G is restrained if and only if 〈x〉 = 〈