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Nearly $$s$$-normal subgroups of a finite group. (English) Zbl 1170.20015
Summary: A subgroup $$H$$ of a group $$G$$ is said to be nearly $$s$$-normal in $$G$$ if there exists a normal subgroup $$N$$ of $$G$$ such that $$HN$$ is a normal subgroup of $$G$$ and $$H\cap N\leq H_{sG}$$, where $$H_{sG}$$ is the largest $$s$$-permutable subgroup of $$G$$ contained in $$H$$. We obtain some results on nearly $$s$$-normal subgroups and supersoluble groups are hence characterized by this kind of subgroups.

##### MSC:
 20D40 Products of subgroups of abstract finite groups 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure 20D25 Special subgroups (Frattini, Fitting, etc.)