Generalized Fitting subgroups of finite groups

TL;DR

This paper explores the properties and influence of generalized Fitting subgroups in finite groups, extending classical concepts like the Fitting subgroup and examining how R-subnormal subgroups affect group structure.

Contribution

It introduces new properties of the generalized Fitting subgroup and analyzes the impact of R-subnormal subgroups on finite group structure.

Findings

Properties of (G)^*(G)(G) are summarized and extended.

The influence of R-subnormal subgroups on finite groups is characterized.

New insights into the structure of finite groups via generalized Fitting subgroups.

Abstract

In this paper we consider the Fitting subgroup $F(G)$ of a finite group $G$ and its generalizations: the quasinilpotent radical $F^*(G)$ and the generalized Fitting subgroup $\tilde{F}(G)$ defined by $\tilde{F}(G)\supseteq \Phi(G)$ and $\tilde{F}(G)/\Phi(G)=Soc(G/\Phi(G))$. We sum up known properties of $\tilde{F}(G)$ and suggest some new ones. Let $R$ be a subgroup of a group $G$. We shall call a subgroup $H$ of $G$ the $R$-subnormal subgroup if $H$ is subnormal in $ \langle H,R\rangle$. In this work the influence of $R$-subnormal subgroups (maximal, Sylow, cyclic primary) on the structure of finite groups are studied in the case when $R\in\{F(G), F^*(G),\tilde{F}(G)\}$.