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# Rendiconti del Seminario Matematico della Università di Padova List online-first RSMUP articles | RSMUP summary
Published online first: 2018-11-20
DOI: 10.4171/RSMUP/13

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

Bin Hu, Jianhong Huang and Alexander N. Skiba

(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)