On \(\tau\)-quasinormal and weakly \(\tau\)-quasinormal subgroups of finite groups.

*(English)*Zbl 1179.20018Summary: 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 \(G\in\mathfrak F\).

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 |