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

Let G be a finite group, and let F be a hereditary saturated formation. We denote by Z(F)(G) the product of all normal subgroups N of G such that every chief factor H/K of G below N is F-central in G, that is, (H/K) (sic) (G/C-G(H/K)) is an element of 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 = A(0) <= A(1) <= ... <= A(n) = G such that either A(i-1) (sic) A(i) or A(i)/(A(i-1))(Ai) is an element of 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 Z(F)(E) = 1 for every subgroup E of G such that S <= E, then C-G (D) <= D, where D = S-F is the F-residual of S.