On the Length of a Maximal Subgroup of a Finite Group

TL;DR

The paper analyzes how a finite group $G$ and its maximal subgroup $M$ constrain several length measures, notably the generalized Fitting height $h^*(G)$ and the non-$p$-soluble length $\lambda_p(G)$. Using a functorial approach based on hereditary Plotkin radicals and Fitting formations, the authors prove universal bounds such as $h^*(G)-h^*(M)\le 2$ and $\lambda_p(G)-\lambda_p(M)\le 1$ under suitable conditions, and show these bounds can be sharpened to $\le 1$ for certain radical constructions. They further demonstrate that for particular $\mathfrak{F}$-radicals, the gaps in $n_\sigma(G,\mathfrak{F})$ relative to maximal subgroups can be made arbitrarily large, providing counterexamples to previously claimed universal bounds. Together, these results map the landscape of possible length differences for maximal subgroups, offering both constructive frameworks and open questions about which formations keep these gaps bounded and by how much.

Abstract

For a finite group $G$ and its maximal subgroup $M$ we proved that the generalized Fitting height of $M$ can't be less by 2 than the generalized Fitting height of $G$ and the non-$p$-soluble length of $M$ can't be less by 1 than the non-$p$-soluble length of $G$. We constructed a hereditary saturated formation $\mathfrak{F}$ such that $\{n_σ(G, \mathfrak{F})-n_σ(M, \mathfrak{F})\mid G$ is finite $σ$-soluble and $M$ is a maximal subgroup of $G\}=\mathbb{N}\cup\{0\}$ where $n_σ(G, \mathfrak{F})$ denotes the $σ$-nilpotent length of the $\mathfrak{F}$-residual of $G$. This construction shows the results about the generalized lengths of maximal subgroups published in Math. Nachr. (1994) and Mathematics (2020) are not correct.

Theorems & Definitions (17)

• Definition 1: Khukhro, Shumyatsky
• Theorem 1
• Definition 2
• Definition 3: Murashka2023
• Theorem 2
• Corollary 2.1
• Corollary 2.2: Doerk1994
• Corollary 2.3: math8122165
• Definition 4
• Theorem 3