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

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

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