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On weakly s-semipermutable or ss-quasinormal subgroups of finite groups

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Abstract

Suppose that G is a finite group and H is a subgroup of G. H is said to be weakly s-semipermutable in G if there are a subnormal subgroup T of G and an s-semipermutable subgroup \(H_{ssG}\) of G contained in H such that \(G=HT\) and \(H\cap T\le H_{ssG}\); H is said to be an ss-quasinormal subgroup of G if there is a subgroup B of G such that \(G=HB\) and H permutes with every Sylow subgroup of B. We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying \(1<|D|<|P|\) and study the structure of G under the assumption that every subgroup H of P with \(|H|=|D|\) is either weakly s-semipermutable or ss-quasinormal in G. Some recent results are generalized and unified.

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Acknowledgements

The paper is dedicated to Professor József Szabados for his 80th birthday. The authors thank the referee for his/her valuable suggestions. It should be said that we could not have polished the final version of this paper well without their outstanding efforts.

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Correspondence to Qingjun Kong.

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The research of the authors is supported by the NNSF of China (11301378) and the Research Grant of Tianjin Polytechnic University.

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Kong, Q., Guo, X. On weakly s-semipermutable or ss-quasinormal subgroups of finite groups. Ricerche mat 68, 571–579 (2019). https://doi.org/10.1007/s11587-018-0427-3

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  • DOI: https://doi.org/10.1007/s11587-018-0427-3

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