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

Abstract

Abstract Let G be a finite group, and let 𝔉 {\mathfrak{F}} be a hereditary saturated formation. We denote by 𝐙 𝔉 ⁢ ( G ) {\mathbf{Z}_{\mathfrak{F}}(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 {\mathfrak{F}} -central in G, that is, ( H / K ) ⋊ ( G / 𝐂 G ⁢ ( H / K ) ) ∈ 𝔉 {(H/K)\rtimes(G/\mathbf{C}_{G}(H/K))\in\mathfrak{F}} . A subgroup A ⩽ G {A\leqslant G} is said to be 𝔉 {\mathfrak{F}} -subnormal in the sense of Kegel, or K- 𝔉 {\mathfrak{F}} -subnormal in G, if there is a subgroup chain A = A 0 ⩽ A 1 ⩽ ⋯ ⩽ A n = G {A=A_{0}\leqslant A_{1}\leqslant\cdots\leqslant A_{n}=G} such that either A i - 1 ⁢ ⊴ ⁢ A i {A_{i-1}\trianglelefteq A_{i}} or A i / ( A i - 1 ) A i ∈ 𝔉 {A_{i}/(A_{i-1})_{A_{i}}\in\mathfrak{F}} for all i = 1 , … , n {i=1,\ldots,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 {\mathfrak{F}} be a hereditary saturated formation containing all nilpotent groups, and let S be a K- F {\mathfrak{F}} -subnormal subgroup of G. If Z F ⁢ ( E ) = 1 {\mathbf{Z}_{\mathfrak{F}}(E)=1} for every subgroup E of G such that S ⩽ E {S\leqslant E} , then C G ⁢ ( D ) ⩽ D {\mathbf{C}_{G}(D)\leqslant D} , where D = S F {D=S^{\mathfrak{F}}} is the F {\mathfrak{F}} -residual of S.