## On weakly $$s$$-permutably embedded subgroups.(English)Zbl 1222.20014

Summary: Let us suppose that $$G$$ 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 $$|H|$$, 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 $$H\cap T\leq H_{se}$$. We investigate the influence of weakly $$s$$-permutably embedded subgroups on the $$p$$-nilpotency and $$p$$-supersolvability of finite groups.

### MSC:

 20D40 Products of subgroups of abstract finite groups 20D35 Subnormal subgroups of abstract finite groups 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks
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