On σ-Residuals of Subgroups of Finite Soluble Groups

Abstract

Let σ={σi:i∈I} be a partition of the set of all prime numbers. A subgroup H of a finite group G is said to be σ-subnormal in G if H can be joined to G by a chain of subgroups H=H0⊆H1⊆⋯⊆Hn=G where, for every j=1,⋯,n, Hj−1 is normal in Hj or Hj/CoreHj(Hj−1) is a σi-group for some i∈I. Let B be a subgroup of a soluble group G normalising the Nσ-residual of every non-σ-subnormal subgroup of G, where Nσ is the saturated formation of all σ-nilpotent groups. We show that B normalises the Nσ-residual of every subgroup of G if G does not have a section that is σ-residually critical.