Subnormality and residuals for saturated formations: A generalization of Schenkman's theorem

Abstract

Let G be a finite group, and let F be a hereditary saturated formation. We denote by ZF(G} the product of all normal subgroups N of G such that every chief factor H/K{H/K} of G below N is F-central in G, that is, (H/K) G/CG(H/K) ϵ F. A subgroup A≤G is said to be F-subnormal in the sense of Kegel, or K-F-subnormal in G, if there is a subgroup chain A=A0≤A1≤ ⋯ ≤An=G such that either Ai-1 Ai or Ai/(Ai-1)Ai ϵ F} for all i=1,⋯,n. In this paper, we prove the following generalization of Schenkman's theorem on the centraliser of the nilpotent residual of a subnormal subgroup: Let F be a hereditary saturated formation containing all nilpotent groups, and let S be a K-F-subnormal subgroup of G. If ZFE=1ZF(E)=1 for every subgroup E of G such that S≤E, then CG(D)≤D, where D=SF is the F-residual of S.