Let G be a finite group. Recall that a subgroup A
of G is said to permute with a subgroup B if AB = BA. A subgroup
A of G is said to be S-quasinormal or S-permutable in G if A
permutes with all Sylow subgroups of G. Recall also that HsG is
the S-permutable closure of H in G, that is, the intersection of all
such S-permutable subgroups of G which contain H. We say that
H is Hall S-permutably embedded in G if H is a Hall subgroup of
the S-permutable closure HsG of H in G.
We prove that the following conditions are equivalent: (1) every
subgroup of G is Hall S-permutably embedded in G; (2) the nilpotent
residual GN of G is a Hall cyclic of square-free order subgroup of
G; (3) G = D ⋊ M is a split extension of a cyclic subgroup D of
square-free order by a nilpotent group M, where M and D are both
Hall subgroups of G.

Description:

Sinitsa D.A. A note on Hall S-permutably embedded subgroups of finite groups / D.A.Sinitsa // Algebra and Discrete Mathematics. - 2017. - Vol. 23. - Number 2. - Рp.305- 311

Koruoğlu, Ö .; Meral, T.; Sahin, R.(ДЗ "ЛНУ імені Тараса Шевченка", 2019)

Let p, q > 2 be relatively prime integers and
let Hp,q be the generalized Hecke group associated to p and q. The
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and Y (z) = −(z + λq) −1 where λp = 2 cos ...

Kurdachenko, L. A.; Subbotin, I. Ya.(Луганский национальный университет им. Т. Шевченко, 2005)

Some properties of abnormal subgroups in generalized soluble groups are considered. In particular, the transitivity of abnormality in metahypercentral groups is proven. Also it is
proven that a subgroup H of a radical ...