Geodesic
A note on Hall $S$-permutably embedded subgroups of finite groups
Algebra and discrete mathematics, Tome 23 (2017) no. 2, pp. 305-311.

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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 $H^{s G}$ 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 $ H^{s G} $ 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 $G^{\mathfrak{N}}$ of $G$ is a Hall cyclic of square-free order subgroup of $G$; (3) $G = D \rtimes M$ 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 $G$.
Keywords: $S$-permutable subgroup, Hall $S$-permutably embedded subgroup, $S$-permutable closure, Sylow subgroup, supersoluble group, maximal subgroup. 
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     author = {Darya A. Sinitsa},
     title = {A note on {Hall}  $S$-permutably embedded subgroups of finite groups},
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     volume = {23},
     number = {2},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2017_23_2_a11/}
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Darya A. Sinitsa. A note on Hall  $S$-permutably embedded subgroups of finite groups. Algebra and discrete mathematics, Tome 23 (2017) no. 2, pp. 305-311. http://geodesic.mathdoc.fr/item/ADM_2017_23_2_a11/

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