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

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

Cite this article

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

DOI 10.4171/RSMUP/13