Finite Groups Having Nonnormal T.I. Subgroups

TL;DR

This paper investigates the structure of finite groups with nonnormal T.I. subgroups that are Hall -subgroups, generalizing Gow's result and establishing conjugacy and normal complement theorems.

Contribution

It proves that Hall T.I. subgroups are conjugate in finite groups and shows that such subgroups are Frobenius complements in -separable groups, extending classical results.

Findings

Hall T.I. subgroups are conjugate in finite groups

Nonnormal T.I. subgroups are Frobenius complements in -separable groups

Generalization of Frobenius's classical normal complement theorem

Abstract

In the present paper, the structure of a finite group $G$ having a nonnormal T.I. subgroup $H$ which is also a Hall $\pi$-subgroup is studied. As a generalization of a result due to Gow, we prove that $H$ is a Frobenius complement whenever $G$ is $\pi$-separable. This is achieved by obtaining the fact that Hall T.I. subgroups are conjugate in a finite group. We also prove two theorems about normal complements one of which generalizes a classical result of Frobenius.