Abstract
Suppose that G is a finite group and H is a subgroup of G. 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; H is said to be weakly s-permutably embedded in G if there are a subnormal subgroup T of G and an s-permutably embedded subgroup \(H_{se}\) of G contained in H such that \(G=HT\) and \(H\cap T\le H_{se}\). 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 ss-quasinormal or weakly s-permutably embedded in G. Some recent results are generalized and unified.
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Acknowledgments
The authors are very grateful to Professor John S. Wilson and the referee for providing valuable suggestions and useful comments, which have greatly improved the final version of the paper. The paper is dedicated to Professor O. H. Kegel for his 80th birthday.
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Communicated by J. S. Wilson.
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 ss-quasinormal or weakly s-permutably embedded subgroups of finite groups. Monatsh Math 182, 637–647 (2017). https://doi.org/10.1007/s00605-016-0877-1
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DOI: https://doi.org/10.1007/s00605-016-0877-1