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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 \(G\in\mathfrak F\).

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