A note on Flavell's theorem associated with Frobenius groups

TL;DR

This paper extends Flavell's theorem on Frobenius groups, demonstrating that the Frobenius kernel is a subgroup even when D is a 2-group, broadening the conditions under which the kernel's subgroup property holds.

Contribution

The paper generalizes Flavell's theorem by proving the Frobenius kernel is a subgroup when D is a 2-group, without relying on character theory.

Findings

K is a subgroup when D is no 2-group

K is a subgroup when D is a 2-group

Extension of Flavell's theorem

Abstract

Let G be a Frobenius group with the Frobenius kernel K. Suppose that G contains a nontrival subgroup D \subseteq K such that the normalizer N_G(D) \not\subseteq K. When D is no 2-group, Flavell proved, without using character theory, that K is a subgroup of G. Based on this result, we further prove that K is a subgroup when D is a 2-group.