Finite solvable groups with a nilpotent normal complement subgroup

TL;DR

This paper investigates the structure of finite solvable groups with specific subgroup conditions, proving the existence of a nilpotent normal complement and describing the group's composition involving Frobenius groups.

Contribution

It establishes conditions under which a non-normal subgroup has a nilpotent normal complement in finite solvable groups, revealing new structural insights.

Findings

Existence of a nilpotent normal complement under certain normalizer conditions

Group decomposes as a product of a nilpotent subgroup and the subgroup H

The complement combined with the Fitting subgroup forms a Frobenius group

Abstract

Let $G$ be a finite solvable group and $H$ a non-normal core-free subgroup of $G$. We show that if the normalizer of any non-trivial normal subgroup of $Fit(H)$ is equal $H$, then $H$ has a nilpotent normal complement $K$ such that $G=KH$ and $KZ(Fit(H))$ is a Frobenius group.