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On $$s$$-semipermutable maximal and minimal subgroups of Sylow $$p$$-subgroups of finite groups. (English) Zbl 1087.20015
Summary: A subgroup $$H$$ of a group $$G$$ is called semipermutable if it is permutable with every subgroup $$K$$ of $$G$$ with $$(|H|,|K|)=1$$. $$H$$ is said to be $$s$$-semipermutable if it is permutable with every Sylow $$p$$-subgroup of $$G$$ with $$(p,|H|)=1$$. In this article, we investigate the $$p$$-nilpotency of a group for which every maximal subgroup of its Sylow $$p$$-subgroups is $$s$$-semipermutable for some prime $$p$$. We generalize some recent theorems by X. Guo and K. P. Shum [Arch. Math. 80, No. 6, 561-569 (2003; Zbl 1050.20010)].

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