Applying the Kövári-Sós-Turán theorem to a question in group theory

Abstract

Let m ≤ n be positive integers and X a class of groups which is closed for subgroups, quotient groups and extensions. Suppose that a finite group G satisfies the condition that for every two subsets M and N of cardinalities m and n, respectively, there exist x ∈ M and y ∈ N such that x, y ∈ X.Let m, n be positive integers and X be a class of groups. We say that a group G satisfies the condition X(m, n) if for every two subsets M and N of cardinalities m and n, respectively, there exist x ∈ M and y ∈ N such that x, y ∈ X. If G satisfies the condition X(m, n), then we write G ∈ X(m, n). In [5] M. Zarrin proposed the following question.