Generalized Fitting subgroups of finite groups

Abstract

In this paper we consider the Fitting subgroup F(G) of a finite group G and its generalizations: the quasinilpotent radical F ∗(G) and the generalized Fitting subgroup F ˜(G) defined by F ˜(G) ⊇ Φ(G) and F ˜(G)/Φ(G) = Soc(G/Φ(G)). We sum up known properties of F ˜(G) and suggest some new ones. Let R be a subgroup of a group G. We shall call a subgroup H of G the R-subnormal subgroup if H is subnormal in hH, Ri. In this work the influence of R-subnormal subgroups (maximal, Sylow, cyclic primary) on the structure of finite groups are studied in the case when R ∈ {F(G), F ∗(G), F ˜(G)}.