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On weakly s-semipermutable subgroups of finite groups. (English) Zbl 1269.20020
Summary: Suppose that $$G$$ is a finite group and $$H$$ is a subgroup of $$G$$. We say that $$H$$ is s-semipermutable in $$G$$ if $$HG_p=G_pH$$ for any Sylow $$p$$-subgroup $$G_p$$ of $$G$$ with $$(p,|H|)=1$$; $$H$$ is weakly s-semipermutable in $$G$$ if there are a subnormal subgroup $$T$$ of $$G$$ and an s-semipermutable subgroup $$H_{ssG}$$ in $$G$$ contained in $$H$$ such that $$G=HT$$ and $$H\cap T\leq H_{ssG}$$. The structure of a finite group with some weakly s-semipermutable subgroups is investigated. Mainly, we get the following local version’s result which is a uniform extension of many recent results in literature:
Main Theorem. Assume that $$p$$ is a fixed prime in $$\pi(G)$$ and $$E$$ is a normal subgroup of $$G$$ and $$Z_{\mathcal U\phi}(G)$$ denotes the product of all normal subgroups $$H$$ of $$G$$ such that all non-Frattini $$p$$-$$G$$-chief factors of $$H$$ have order $$p$$. Then $$E\leq Z_{\mathcal U_p\phi}(G)$$ if there exists a normal subgroup $$X$$ of $$G$$ such that $$F^*_p(E)\leq X\leq E$$, where $$F^*_p(E)$$ is the generalized $$p$$-Fitting subgroup of $$E$$, and $$X$$ satisfies the following: for any Sylow $$p$$-subgroup $$P$$ of $$X$$, $$P$$ has a subgroup $$D$$ such that $$1<|D|<|P|$$ and all subgroups $$H$$ of $$P$$ with order $$|H|=|D|$$ and all cyclic subgroups of $$P$$ with order 4 (if $$P$$ is a non-Abelian 2-group and $$|D|=2$$) are weakly s-semipermutable in $$G$$.

##### MSC:
 20D40 Products of 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 20D15 Finite nilpotent groups, $$p$$-groups 20D25 Special subgroups (Frattini, Fitting, etc.) 20D35 Subnormal subgroups of abstract finite groups
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