Thompson-like characterization of solubility for products of finite groups

Abstract

A remarkable result of Thompson states that a finite group is soluble if and only if all its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory of groups, aiming for global properties of groups from local properties of two-generated (or more generally, n-generated) subgroups. We contribute an extension of Thompson's theorem from the perspective of factorized groups. More precisely, we study finite groups G = AB with subgroups A, B such that a, b is soluble for all a ∈ A and b ∈ B. In this case, the group G is said to be an S-connected product of the subgroups A and B for the class S of all finite soluble groups. Our main theorem states that G = AB is S-connected if and only if [A, B] is soluble. In the course of the proof we derive a result about independent primes regarding the soluble graph of almost simple groups that might be interesting in its own right.

Products of Finite Connected Subgroups

“…Again, the application of Theorem 1, with A = G, and B = x ≤ G, assures that x G S is a normal (soluble) subgroup of G under the hypothesis in statement (1) and, therefore, Theorem 3 is also recovered. It is to be emphasized that we make use of this result in the proof of our Theorem 1.…”

“…We extend previous research on the influence of two-generated subgroups on the structure of groups, in connection with the study of products of subgroups. In [1], the following result is proven: Theorem 1. Let the finite group G = AB be the product of subgroups A and B.…”