On σ -Residuals of Subgroups of Finite Soluble Groups

Let σ = { σ i : 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 = H 0 ⊆ H 1 ⊆ ⋯ ⊆ H n = G where, for every j = 1 , ⋯ , n , H j − 1 is normal in H j or H j / C o r e H j ( H j − 1 ) is a σ i -group for some i ∈ I . Let B be a subgroup of a soluble group G normalising the N σ -residual of every non- σ -subnormal subgroup of G , where N σ is the saturated formation of all σ -nilpotent groups. We show that B normalises the N σ -residual of every subgroup of G if G does not have a section that is σ -residually critical.