Invariant TI-subgroups and structure of finite groups

Abstract

Let G be a finite group and assume that a group of automorphisms A is acting on G such that A and G have coprime orders. Recall that a subgroup H of G is said to be a TI-subgroup if it has trivial intersection with its distinct conjugates in G . We study the solubility and other properties of G when we assume that certain invariant subgroups of G are TI-subgroups, precisely when all A -invariant subgroups, all non-nilpotent A -invariant subgroups, and all non-abelian A -invariant subgroups of G , respectively, are TI-subgroups.