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Published online first: 2018-11-20
DOI: 10.4171/RSMUP/13

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

Bin Hu[1], Jianhong Huang[2] and Alexander N. Skiba[3]

(1) Jiangsu Normal University, Xuzhou, Jiangsu, China
(2) Jiangsu Normal University, Xuzhou, Jiangsu, China
(3) Francisk Skorina Gomel State University, Belarus

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 $(H/K) \rtimes (G/C_{G}(H/K))$ 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 $F^{*}_{\sigma}(G)$). Our main goal here is to study the relations between the subgroups $F_{\sigma}(G)$ and $F^{*}_{\sigma}(G)$, and the influence of these two subgroups on the structure of $G$.

Keywords: Finite group, $\sigma$-nilpotent group, $\sigma$-quasinilpotent group, $\sigma$-Fitting subgroup, generalized $\sigma$-Fitting subgroup

Hu Bin, Huang Jianhong, Skiba Alexander: On the generalized $\sigma$-Fitting subgroup of finite groups. Rend. Sem. Mat. Univ. Padova Electronically published on November 20, 2018. doi: 10.4171/RSMUP/13 (to appear in print)