On the generalized $\sigma$-Fitting subgroup of finite groups

Abstract

Let $\sigma =\{{\sigma }_i|i\in I\}$ be some partition of the set $\mathbb{P}$ of all primes, and let $G$ be a finite group. A chief factor $H/K$ of $G$ is said to be $\sigma$-central (in $G$) if the semidirect product $(H/K)⋊(G/C_G(H/K))$ is a ${\sigma }_i$-group for some $i=i(H/K)$; otherwise, it is called $\sigma$-eccentric (in $G$). We say that $G$ is: $\sigma$-nilpotent if every chief factor of $G$ is $\sigma$-central; $\sigma$-quasinilpotent if for every $\sigma$-eccentric chief factor $H/K$ of $G$, every automorphism of $H/K$ induced by an element of $G$ is inner. The product of all normal $\sigma$-nilpotent (respectively $\sigma$-quasinilpotent) subgroups of $G$ is said to be the $\sigma$-Fitting subgroup (respectively the generalized $\sigma$-Fitting subgroup) of $G$ and we denote it by $F_{\sigma }(G)$ (respectively by $F_{\sigma }^{\ast }(G)$). Our main goal here is to study the relations between the subgroups $F_{\sigma }(G)$ and $F_{\sigma }^{\ast }(G)$, and the influence of these two subgroups on the structure of $G$.