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 ) \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-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-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 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 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 \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-residual of 𝑆.

Finite groups with given sets of 𝔉-subnormal subgroups

In this paper, the classes of groups with given systems of [Formula: see text]-subnormal subgroups are studied. In particular, it is showed that if [Formula: see text] and [Formula: see text] are a saturated homomorph and a hereditary saturated formation, respectively, then the class of groups whose [Formula: see text]-subgroups are all [Formula: see text]-subnormal is a hereditary saturated formation. As corollaries, some known results about supersoluble groups, classes of groups with [Formula: see text]-subnormal cyclic primary and Sylow subgroups are obtained. Also the new characterization of the class of groups whose extreme subgroups all belong [Formula: see text], where [Formula: see text] is a hereditary saturated formation, is obtained.