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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References
Ballester-Bolinches, A., Ezquerro, L.M., Skiba, A.N.: Local embeddings of some families of subgroups of finite group. Acta Math. Sin. (Engl. Ser.) 25, 869–882 (2009)
Deskins, W.E.: On quasinormal subgroups of finite groups. Math. Z. 82, 125–132 (1963)
Huppert, B.: Endliche Gruppen I. Springer, Berlin (1967)
Huppert, B., Blackburn, N.: Finite Groups III. Springer, Berlin (1982)
Kegel, O.H.: Sylow Gruppen und subnormalteiler endlicher Gruppen. Math. Z. 78, 205–221 (1962)
Li, S., Shen, Z., Kong, X.: on \(ss\)-quasinormal subgroups of finite subgroups. Commun. Algebra 36, 4436–4447 (2008)
Li, S., Shen, Z., Liu, J., et al.: The influence of \(ss\)-quasinormality of some subgroups on the structure of finite groups. J. Algebra 319, 4275–4287 (2008)
Li, Y., Qiao, S., Su, N., et al.: On weakly \(s\)-semipermutable subgroups of finite groups. J. Algebra 371, 250–261 (2012)
Li, Y., Wei, H., Wang, Y.: The influence of \(\pi \)-quasinormality of some subgroups of a finite group. Arch. Math. 81, 245–252 (2003)
Li, Y., Wang, Y.: The influence of minimal subgroups on the structure of a finite group. Proc. Am. Math. Soc. 131, 337–341 (2002)
Skiba, A.N.: On weakly \(s\)-permutable subgroups of finite groups. J. Algebra 315, 192–209 (2007)
Schmid, P.: Subgroups permutable with all Sylow subgroups. J. Algebra 207, 285–293 (1998)
Wei, X., Guo, X.: On \(ss\)-quasinormal subgroups and the structure of finite groups. Sci. China Math. 54(3), 449–456 (2011)
Zhang, Q., Wang, L.: The influence of \(s\)-semipermutable subgroups on the structure of a finite group. Acta Math. Sin. 48, 81–88 (2005)
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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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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