Abstract:
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 $G$.
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