Length parameters of finite groups and their Hall subgroups

TL;DR

This paper establishes bounds on the non-$p$-soluble length of finite groups based on properties of their Hall subgroups, using classification results and properties of simple groups.

Contribution

It introduces new bounds relating the non-$p$-soluble length of a finite group to the generalized Fitting height and 2-length of its Hall subgroups, extending previous classification-based results.

Findings

Non-$p$-soluble length is bounded by the generalized Fitting height of Hall $ ext{ extpi}$-subgroups.

If the Hall subgroup is soluble, the bound involves twice its 2-length plus one.

Finite simple groups of order divisible by $p$ cannot have nilpotent Hall $ ext{ extlbrack}2,p extbrack}$-subgroups.

Abstract

Let $\pi$ be a set of primes containing $2$ and an odd prime $p$. It is proved that if a finite group $G$ has a Hall $\pi$-subgroup $H$, then the non-$p$-soluble length of $G$ is bounded above by the generalized Fitting height of $H$. The proof uses the fact, obtained in [4] using the classification of finite simple groups, that a finite simple group of order divisible by $p$ cannot have a nilpotent Hall $\{2,p\}$-subgroup. As a corollary, it is proved that if in addition $H$ is soluble, then the non-$p$-soluble length of $G$ is bounded above by $2l_2(H)+1$, where $l_2(H)$ is the $2$-length of $H$.