On Maximal Subgroups of a Finite Solvable Group

TL;DR

This paper investigates the structure of maximal subgroups in finite solvable groups, establishing conditions under which certain Sylow subgroups are contained within these maximal subgroups.

Contribution

It provides a new result linking non-normal maximal subgroups and Sylow subgroups in finite solvable groups.

Findings

Existence of Sylow q-subgroups with specific normalizer properties

Conditions relating the Fitting subgroup and maximal subgroups

Structural insights into maximal subgroup configurations

Abstract

The following result is received: Let $H$ be a non-normal maximal subgroup of a finite solvable group $G$ and let $q \in \pi(F(H/\mathrm{Core}_GH))$, then $G$ has a Sylow $q$-subgroup $Q$ such that $N_{G}(Q) \subseteq H$.