Variations on the Thompson theorem

Abstract

Thompson's theorem states that a finite group G is solvable if and only if every 2-generated subgroup of G is solvable. In this paper, we prove some new criteria for both solvability and nilpotency of a finite group using certain conditions on 2-generated subgroups. We show that a finite group G is solvable if and only if for every pair of two elements x and y in G of coprime prime power order, if $〈x,y〉$ is solvable, then $〈x,y^g〉$ is solvable for all $g\in G$. Similarly, a finite group G is nilpotent if and only if for every pair of elements x and y in G of coprime prime power order, if $〈x,y〉$ is solvable, then x and $y^g$ commute for some $g\in G$. Some applications to graphs defined on groups are given.