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On $$\tau$$-quasinormal and weakly $$\tau$$-quasinormal subgroups of finite groups. (English) Zbl 1179.20018
Summary: Let $$G$$ be a finite group and $$H$$ a subgroup of $$G$$. We put $$\tau_G(H)=\{q\in\pi(G)\setminus\pi(H)\mid (|H|,|Q^G|)\neq 1$$ for a Sylow $$q$$-subgroup $$Q$$ of $$G\}$$. We say that: (1) $$H$$ is $$\tau$$-quasinormal in $$G$$ if $$H$$ commutes with all Sylow $$q$$-subgroups of $$G$$ for all $$q\in\tau_G(H)$$; (2) $$H$$ is weakly $$\tau$$-quasinormal in $$G$$ if $$G$$ has a subnormal subgroup $$T$$ such that $$HT=G$$ and $$T\cap H\leq H_{\tau G}$$, where $$H_{\tau G}$$ is the subgroup generated by all those subgroups of $$H$$ which are $$\tau$$-quasinormal in $$G$$.
Our main result here is the following Theorem 1.3. Let $$\mathfrak F$$ be a saturated formation containing all supersoluble groups and $$G$$ a group with a normal subgroup $$E$$ such that $$G/E\in\mathfrak F$$. Suppose that every non-cyclic Sylow subgroup $$P$$ of $$E$$ has a subgroup $$D$$ such that $$1<|D|<|P|$$ and every subgroup $$H$$ of $$P$$ with order $$|H|=|D|$$ and every cyclic subgroup of $$P$$ with order 4 (if $$|D|=2$$ and $$P$$ is a non-Abelian 2-group) not having a supersoluble supplement in $$G$$ are weakly $$\tau$$-quasinormal in $$G$$. Then $$G\in\mathfrak F$$.

##### MSC:
 20D40 Products of subgroups of abstract finite groups 20D35 Subnormal subgroups of abstract finite groups 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks