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On weakly τ-quasinormal subgroups of finite groups

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Abstract

Let G be a finite 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 τG , where H τG 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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  1. Department of Mathematics, Gomel Francisk Skorina State University, Gomel, 246019, Belarus

    V. O. Lukyanenko & A. N. Skiba

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  1. V. O. Lukyanenko
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  2. A. N. Skiba
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Correspondence to V. O. Lukyanenko.

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Lukyanenko, V.O., Skiba, A.N. On weakly τ-quasinormal subgroups of finite groups. Acta Math Hung 125, 237–248 (2009). https://doi.org/10.1007/s10474-009-9008-y

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