On the generalized σ-Fitting subgroup of finite groups

Abstract

Let σ = {σi|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) o (G/CG(H/K)) is a σi-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 Fσ(G) (respectively by Fσ*(G)). Our main goal here is to study the relations between the subgroups Fσ(G) and Fσ*(G), and the influence of these two subgroups on the structure of G.