On weakly τ-quasinormal subgroups of finite groups (original) (raw)
2015, Acta Mathematica Hungarica
Let G be a nite group and H a subgroup of G. We say that: (1) H is τ-quasinormal in G if H permutes with all Sylow subgroups Q of G such that |Q|, |H| = 1 and |H|, |Q G | = 1; (2) H is weakly τ-quasinormal in G if G has a subnormal subgroup T such that HT = G and T ∩ H H τ G , where H τ G is the subgroup generated by all those subgroups of H which are τ-quasinormal in G. Our main result here is the following. Let F be a saturated formation containing all supersoluble groups and let X E be normal subgroups of a group G such that G/E ∈ F. Suppose that every non-cyclic Sylow subgroup P of X 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 non-Abelian) not having a supersoluble supplement in G is weakly τ-quasinormal in G. If X is either E or F * (E), then G ∈ F.
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