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On Finite NDC-Groups

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Abstract

A finite p-group G is called a NDC-group if \(G'/{G' \cap N}\) is cyclic or \(N/{N \cap Z(G)}\) is cyclic for every normal subgroup N in G. In this paper, first we give some properties of NDC-groups and in particular, we give the order of \(G'Z(G)/Z(G)\) for a NDC-group G so that we could investigate the structure of NDC-groups. Next, some necessary and sufficient conditions for a p-group G whose quotient group G / Z(G) is generated by two elements to be a NDC-group are given. Finally, we also give some further properties of p-groups with small derived subgroups by using the properties of capable p-groups with small generating systems, which maybe could apply some other research of p-groups.

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References

  1. An, L.J., Qu, H.P., Xu, M.Y., Yang, C.S.: Quasi-\(NC\)-groups. Commun. Algebra 36, 4011–4019 (2008)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Bai, P.F., Guo, X.Y.: On finite \(p\)-groups with few normal subgroups. J. Algebra Appl. 16, 1750159 (2017)

    Article  MathSciNet  Google Scholar 

  4. Bannuscher, W.: Über \(k\)-\(\Phi \)-reguläre \(p\)-Gruppen. Math. Nachr. 137, 7–17 (1988)

    Article  MathSciNet  Google Scholar 

  5. Berkovich, Y.: Groups of Prime Power Order. Vol. 1. De Gruyter Expositions in Mathematics, vol. 46. Walter de Gruyter GmbH & Co. KG, Berlin (2008)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Blackburn, N.: Generalizations of certain elementary theorems on \(p\)-groups. Proc. Lond. Math. Soc. 11, 1–22 (1961)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Hall, M., Senior, J.K.: The Groups of Order \(2^n\) (\(n \le 6\)). MacMillan, New York (1964)

    MATH  Google Scholar 

  10. Huppert, B.: Endliche Gruppen I. Die Grundlehren der Mathematischen Wissenschaften, vol. 134. Springer, Berlin (1967)

    Google Scholar 

  11. Huppert, B., Blackburn, N.: Finite Groups II. Grundlehren der Mathematischen Wissenschaften, vol. 242. Springer, Berlin (1982)

    MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  13. Li, L.L., Qu, H.P., Chen, G.Y.: Central extension of inner abelian \(p\)-groups I (Chinese). Acta Math. Sin. 53, 675–684 (2010)

    MATH  Google Scholar 

  14. Nagrebeckii, V.T.: Invariant coverings of subgroups. Ural. Gos. Univ. Mat. Zap. 5, 91–100 (1966)

    MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  16. Nagrebeckii, V.T.: Finite groups in which any non-nilpotent subgroup is invariant. Ural. Gos. Univ. Mat. Zap. 6, 45–49 (1968)

    MathSciNet  MATH  Google Scholar 

  17. Qu, H.P., Hu, R.F.: Central extension of inner abelian \(p\)-groups III (Chinese). Acta Math. Sin. 53, 1051–1064 (2010)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Romalis, G.M., Sesekin, N.F.: Metahamiltonian groups. Ural. Gos. Univ. Mat. Zap. 5, 101–106 (1966)

    MathSciNet  Google Scholar 

  20. Romalis, G.M., Sesekin, N.F.: Metahamiltonian groups II. Ural. Gos. Univ. Mat. Zap. 6, 50–52 (1968)

    MathSciNet  MATH  Google Scholar 

  21. Romalis, G.M., Sesekin, N.F.: Metahamiltonian groups III. Ural. Gos. Univ. Mat. Zap. 7, 195–199 (1969–1970)

  22. Shahriari, S.: On normal subgroups of capable groups. Arch. Math. (Basel) 48, 193–198 (1987)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  24. Silberberg, G.: Finite \(p\)-groups with few normal subgroups. Bull. Belg. Math. Soc. 7, 365–376 (2000)

    Article  MathSciNet  Google Scholar 

  25. Silberberg, G.: Finite \(p\)-groups with few normal subgroups(II). Bull. Belg. Math. Soc. 9, 185–192 (2002)

    Article  MathSciNet  Google Scholar 

  26. Xu, M.Y.: A theorem on metabelian \(p\)-groups and some consequences. Chin. Ann. Math. 5B, 1–6 (1984)

    MathSciNet  MATH  Google Scholar 

  27. 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 

  28. Zhang, L.H., Zhang, J.Q.: Finite \(p\)-groups all of whose non-normal abelian subgroups are cyclic. J. Algebra Appl. 12, 1350052 (2013)

    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, X.H., Guo, X.Y.: Finite \(p\)-groups whose non-normal cyclic subgroups have small index in their normalizers. J. Group Theory 15, 641–659 (2012)

    MathSciNet  MATH  Google Scholar 

  31. Zhao, L.B., Guo, X.Y.: Finite \(p\)-groups in which the normal closures of the non-normal cyclic subgroups have small index. J. Algebra Appl. 13, 1350087 (2014)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank the referees for their valuable suggestions and useful comments contributed to the final version of this paper.

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Authors and Affiliations

  1. School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, 030006, People’s Republic of China

    Pengfei Bai

  2. Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

    Xiuyun Guo

  3. Institute of Mathematics, Yunnan University, Kunming, 650091, People’s Republic of China

    K. P. Shum

Authors
  1. Pengfei Bai
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  2. Xiuyun Guo
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  3. K. P. Shum
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Correspondence to Xiuyun Guo.

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Communicated by V. Ravichandran.

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The research of the work was partially supported by the National Natural Science Foundation of China (11771271, 11801334).

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Bai, P., Guo, X. & Shum, K.P. On Finite NDC-Groups. Bull. Malays. Math. Sci. Soc. 43, 3099–3123 (2020). https://doi.org/10.1007/s40840-019-00856-z

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  • DOI: https://doi.org/10.1007/s40840-019-00856-z

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