# On the generalized $σ$-Fitting subgroup of finite groups

### Bin Hu

Jiangsu Normal University, Xuzhou, Jiangsu, China### Jianhong Huang

Jiangsu Normal University, Xuzhou, Jiangsu, China### Alexander N. Skiba

Francisk Skorina Gomel State University, Belarus

## Abstract

Let $σ={σ_{i}∣i∈I}$ be some partition of the set $P$ of all primes, and let $G$ be a finite group. A chief factor $H/K$ of $G$ is said to be $σ$-*central* (in $G$) if the semidirect product $(H/K)⋊(G/C_{G}(H/K))$ is a $σ_{i}$-group for some $i=i(H/K)$; otherwise, it is called $σ$-*eccentric* (in $G$). We say that $G$ is: $σ$-*nilpotent* if every chief factor of $G$ is $σ$-central; $σ$-*quasinilpotent* if for every $σ$-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 $σ$-nilpotent (respectively $σ$-quasinilpotent) subgroups of $G$ is said to be the $σ$-*Fitting subgroup* (respectively the *generalized* $σ$-*Fitting subgroup*) of $G$ and we denote it by $F_{σ}(G)$ (respectively by $F_{σ}(G)$). Our main goal here is to study the relations between the subgroups $F_{σ}(G)$ and $F_{σ}(G)$, and the influence of these two subgroups on the structure of $G$.

## Cite this article

Bin Hu, Jianhong Huang, Alexander N. Skiba, On the generalized $σ$-Fitting subgroup of finite groups. Rend. Sem. Mat. Univ. Padova 141 (2019), pp. 19–36

DOI 10.4171/RSMUP/13