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A Classification of Finite Metahamiltonian p-Groups

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Abstract

A finite non-abelian group G is called metahamiltonian if every subgroup of G is either abelian or normal in G. If G is non-nilpotent, then the structure of G has been determined. If G is nilpotent, then the structure of G is determined by the structure of its Sylow subgroups. However, the classification of finite metahamiltonian p-groups is an unsolved problem. In this paper, finite metahamiltonian p-groups are completely classified up to isomorphism.

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References

  1. An, L.J.: Finite \(p\)-groups whose non-normal subgroups have few orders. Front. Math. China 13(4), 763–777 (2018)

    Article  MathSciNet  Google Scholar 

  2. An, L.J., Hu, R.F., Zhang, Q.H.: Finite \(p\)-groups with a minimal non-abelian subgroup of index \(p\) (IV). J. Algebra Appl. 14(2), 1550020 (2015). (54 pages)

    Article  MathSciNet  Google Scholar 

  3. An, L.J., Li, L.L., Qu, H.P., Zhang, Q.H.: Finite \(p\)-groups with a minimal non-abelian subgroup of index \(p\) (II). Sci. China Math. 57, 737–753 (2014)

    Article  MathSciNet  Google Scholar 

  4. An, L.J., Peng, J.: Finite \(p\)-groups in which any two noncommutative elements generate an inner abelian group of order \(p^4\). Algebra Colloq. 20, 215–226 (2013)

    Article  MathSciNet  Google Scholar 

  5. An, L.J., Zhang, Q.H.: Finite metahamilton \(p\)-groups. J. Algebra 442, 23–35 (2015)

    Article  MathSciNet  Google Scholar 

  6. Berkovich, Y.: Finite \(p\)-groups with few minimal nonabelian subgroups. J. Algebra 297, 62–100 (2006)

    Article  MathSciNet  Google Scholar 

  7. Berkovich, Y., Janko, Z.: Structure of finite \(p\)-groups with given subgroups. Contemp. Math. 402, 13–93 (2006)

    Article  MathSciNet  Google Scholar 

  8. Berkovich, Y., Janko, Z.: Groups of Prime Power Order, vol. 2. Walter de Gruyter, Berlin (2008)

    Book  Google Scholar 

  9. Blackburn, N.: On prime power groups with two generators. Proc. Camb. Philos. Soc. 54, 327–337 (1957)

    Article  MathSciNet  Google Scholar 

  10. Dedekind, R.: Über Gruppen, deren samtliche Teiler Normalteiler sind. Math. Ann. 48, 548–561 (1897)

  11. Draganyuk, S.V.: On the structure of finite primary groups all 2-maximal subgroups of which are abelian (Russian). In: Complex Analysis, Algebra and Topology. Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, pp. 42–51 (1990)

  12. Kazarin, L.S.: On certain classes of finite groups. Dokl. Akad. Nauk SSSR (Russ.) 197, 773–776 (1971)

    MathSciNet  Google Scholar 

  13. King, B.W.: Presentations of metacyclic groups. Bull. Austral. Math. Soc. 8, 103–131 (1973)

    Article  MathSciNet  Google Scholar 

  14. Nagrebeckii, V.T.: Finite non-nilpotent groups any non-abelian subgroup of which is normal. Ural. Gos. Univ. Mat. Zap. 6, 80–88 (1967)

    MathSciNet  MATH  Google Scholar 

  15. Newman, M.F., Xu, M.Y.: Metacyclic groups of prime-power order (research announcement). Adv. Math. (Beijing) 17, 106–107 (1988)

    Google Scholar 

  16. Passman, D.S.: Nonnormal subgroups of \(p\)-groups. J. Algebra 15, 352–370 (1970)

    Article  MathSciNet  Google Scholar 

  17. Qu, H.P., Yang, S.S., Xu, M.Y., An, L.J.: Finite \(p\)-groups with a minimal non-abelian subgroup of index \(p\) (I). J. Algebra 358, 178–188 (2012)

    Article  MathSciNet  Google Scholar 

  18. Qu, H.P., Xu, M.Y., An, L.J.: Finite \(p\)-groups with a minimal non-abelian subgroup of index \(p\) (III). Sci. China Math. 58, 763–780 (2015)

    Article  MathSciNet  Google Scholar 

  19. Qu, H.P., Zhao, L.P., Gao, J., An, L.J.: Finite \(p\)-groups with a minimal non-abelian subgroup of index \(p\) (V). J. Algebra Appl. 13(7), 1450032 (2014). (35 pages)

    Article  MathSciNet  Google Scholar 

  20. Rédei, L.: Das “schiefe Produkt” in der Gruppentheorie mit Anwendung auf die endlichen nichtkommutativen Gruppen mit lauter kommutativen echten Untergruppen und die Ordnungszahlen, zu denen nur kommutative Gruppen gehören(German). Comment. Math. Helvet. 20, 225–264 (1947)

    Article  Google Scholar 

  21. Sheriev, V.A.: A description of the class of finite p-groups whose 2-maximal subgroups are all abelian II. Primary groups. Proc. Sem. Algebr. Syst. 2, 54–76 (1970)

    Google Scholar 

  22. Song, Q.W., Qu, H.P.: Finite \(p\)-groups whose nonnormal subgroups are metacyclic. Sci. China Math. 63(7), 1271–1284 (2020)

    Article  MathSciNet  Google Scholar 

  23. Song, Q.W., Zhang, Q.H.: Finite \(2\)-groups whose length of nonnormal subgroups is at most 2. Front. Math. China 13(5), 1075–1097 (2018)

    Article  MathSciNet  Google Scholar 

  24. Xu, M.Y., An, L.J., Zhang, Q.H.: Finite \(p\)-groups all of whose non-abelian proper subgroups are generated by two elements. J. Algebra 319, 3603–3620 (2008)

    Article  MathSciNet  Google Scholar 

  25. Xu, M.Y., Zhang, Q.H.: A classification of metacyclic \(2\)-groups. Algebra Colloq. 13, 25–34 (2006)

    Article  MathSciNet  Google Scholar 

  26. Zhang, Q.H., Guo, X.Q., Qu, H.P., Xu, M.Y.: Finite groups which have many normal subgroups. J. Korean Math. Soc. 46, 1165–1178 (2009)

    Article  MathSciNet  Google Scholar 

  27. Zhang, Q.H., Li, X.X., Su, M.J.: Finite \(p\)-groups whose nonnormal subgroups have orders at most \(p^3\). Front. Math. China. 9(5), 1169–1194 (2014)

    Article  MathSciNet  Google Scholar 

  28. Zhang, Q.H., Su, M.J.: Finite \(2\)-groups whose nonnormal subgroups have orders at most \(2^3\). Front. Math. China 7(5), 971–1003 (2012)

    Article  MathSciNet  Google Scholar 

  29. Zhang, Q.H., Sun, X.J., An, L.J., Xu, M.Y.: Finite \(p\)-groups all of whose subgroups of index \(p^2\) are abelian. Algebra Colloq. 15, 167–180 (2008)

    Article  MathSciNet  Google Scholar 

  30. Zhang, Q.H., Zhao, L.B., Li, M.M., Shen, Y.Q.: Finite \(p\)-groups all of whose subgroups of index \(p^3\) are abelian. Commun. Math. Stat. 3(1), 69–162 (2015)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Thanks to the referee(s) for his or her careful reading and valuable comments. These improve the whole paper.

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Authors and Affiliations

  1. School of Mathematics Science, Peking University, Beijing, 100087, People’s Republic of China

    Xingui Fang

  2. Department of Mathematics, Shanxi Normal University, Linfen, 041004, Shanxi, People’s Republic of China

    Lijian An

Authors
  1. Xingui Fang
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  2. Lijian An
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Correspondence to Lijian An.

Additional information

This work was supported by NSFC (Nos. 11971280,11771258).

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Fang, X., An, L. A Classification of Finite Metahamiltonian p-Groups. Commun. Math. Stat. 9, 239–260 (2021). https://doi.org/10.1007/s40304-020-00229-0

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  • DOI: https://doi.org/10.1007/s40304-020-00229-0

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