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Suppose G is a finite group and H is a subgroup of G. H is called weakly s-permutably embedded in G if there are a subnormal subgroup T of G and an s-permutably embedded subgroup $H_{se}$ of G contained in H such that G = HT and $H\cap T\le H_{se}$; H is called weakly s-supplemented in G if there is a subgroup T of G such that G = HT and $H\cap T\le H_{sG}$, where $H_{sG}$ is the subgroup of H generated by all those subgroups of H which are s-permutable in G. We investigate the influence of the existence of s-permutably embedded and weakly s-supplemented...

Suppose $G$ is a finite group and $H$ is a subgroup of $G$. $H$ is said to be $s$-permutably embedded in $G$ if for each prime $p$ dividing $\left|H\right|$, a Sylow $p$-subgroup of $H$ is also a Sylow $p$-subgroup of some $s$-permutable subgroup of $G$; $H$ is called weakly $s$-permutably embedded in $G$ if there are a subnormal subgroup $T$ of $G$ and an $s$-permutably embedded subgroup $H_{se}$ of $G$ contained in $H$ such that $G=HT$ and $H\cap T\le H_{se}$. We investigate the influence of weakly $s$-permutably embedded subgroups on the $p$-nilpotency and $p$-supersolvability of finite...

The major aim of the present paper is to strengthen a nice result of Shemetkov and Skiba which gives some conditions under which every non-Frattini G-chief factor of a normal subgroup E of a finite group G is cyclic. As applications, some recent known results are generalized and unified.