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Finite 2-Groups Whose Number of Subgroups of Each Order are at Most \(2^4\)

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Abstract

Assume G is a group of order \(2^n, n\ge 5\). Let \(s_k(G)\) denote the number of subgroups of order \(2^k\) of G. We classify finite 2-groups G with \(s_k(G)\le 2^4,\) where \(1\le k\le n\).

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References

  1. An, L.J., Liu, Y.D.: Describing finite \(p\)-groups with the number of subgroups. Adv. Math. (China) 40(3), 285–292 (2011)

    MathSciNet  Google Scholar 

  2. Berkovich, Y.: Groups of Prime Power Order, vol. 1. Walter de Gruyter, Berlin (2008)

    MATH  Google Scholar 

  3. Berkovich, Y.: On the number of subgroups of given order in a finite \(p\)-group of exponent \(p\). Proc. Am. Math. Soc. 109, 875–879 (1990)

    MathSciNet  MATH  Google Scholar 

  4. Besche, H.U., Eick, B., O’Brien, E.A.: A millennium project: constructing small groups. Int. J. Algebra Comput. 12, 623–644 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bosma, W., Cannon, J., Playoust, C.: The MAGMA algebra system I: the user language. J. Symb. Comput. 24, 235–265 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Burnside, W.: Theory of Groups of Finite Order. Cambridge University Press, Cambridge (1897)

    MATH  Google Scholar 

  7. Dyubyuk, P.E.: On the number of subgroups of certain categories of finite \(p\)-groups. Mat. Sbornik N.S. 30(72), 575–580 (1952) (Russian)

  8. Fan, Y.: A characterization of elementary abelian \(p\)-groups by counting subgroups. Math. Pract. Theory 1, 63–64 (1988) (in Chinese)

  9. Hua, L.K., Tuan, H.F.: Determination of the groups of odd-prime-power order \(p^n\) which contain a cyclic subgroup of index \(p^2\). Sci. Rep. Natl. Tsing. Hua Univ. Ser. A 4, 145–154 (1940)

    Google Scholar 

  10. Hua, L.K.: Some “Anzahl” theorems for groups of prime power orders. Sci. Rep. Natl. Tsing Hua Univ. 4, 313–327 (1947)

    MathSciNet  MATH  Google Scholar 

  11. Kulakoff, A.: Über die Anzahl der eigentlichen Untergruppen und der Elemente von gegebener Ordnung in \(p\)-Gruppen. Math. Ann. 104, 778–793 (1931)

    Article  MathSciNet  MATH  Google Scholar 

  12. Li, L.L., Qu, H.P., Chen, G.Y.: Central extension of inner abelian \(p\)-groups. I. Acta Math. Sin. (Chin. Ser.) 53:4, 675–684 (2010) (Chinese)

  13. Li, L.L., Qu, H.P.: The number of conjugacy classes of nonnormal subgroups of finite \(p\)-groups. J. Algebra 466, 44–62 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. McKelden, A.M.: Groups of order \(2^m\) that contain cyclic subgroups of order \(2^{m-3}\). Am. Math. Monthly 13(6–7), 121–136d (1906)

    MathSciNet  Google Scholar 

  15. Ninomiya, Y.: Finite \(p\)-groups with cyclic subgroups of index \(p^2\). Math. J. Okayama Univ. 36, 1–21 (1994)

    MathSciNet  MATH  Google Scholar 

  16. Qu, H.P., Song, Y., Zhang, Q.H.: Finite \(p\)-groups in which the number of subgroups of possible order are less than or equal \(p^3 \). Chin. Ann. Math. 31B(4), 497–506 (2010)

    Article  MATH  Google Scholar 

  17. Qu, H.P.: Finite non-elementary abelian \(p\)-groups whose number of subgroups is maximal. Israel J. Math. 195(2), 773–781 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Rédei, L.: Das schiefe Produkt in der gruppentheorie. Comment. Math. Helvet. 20, 225–267 (1947)

    Article  MATH  Google Scholar 

  19. Tuan, H.F.: An Anzahl theorem of Kulakoff’s type for \(p\)-groups. Sci. Rep. Natl. Tsing Hua Univ. Ser. A 5, 182–189 (1948)

    MathSciNet  MATH  Google Scholar 

  20. Wang, L.F.: Finite \(p\)-groups whose number of subgroups of each order is at most \(p^4\), Algebra Colloquium (submitted)

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang, Q.H.: A characterization of metacyclic p-groups by counting subgroups. In: Proceedings of the International Conference on Algebra, 2010, pp. 713–720, World Science Publication, Hackensack, NJ (2012)

  24. Zhang, Q.H., Li, P.J.: Finite \(p\)-groups with a cyclic subgroup of index \(p^3\). J. Math. Res. Appl. 32(5), 505–529 (2012)

    MathSciNet  MATH  Google Scholar 

  25. Zhang, Q.H., Qu, H.P.: On Hua-Tuan’s conjecture. Sci. China Ser. A 52:2, 389–393 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhang, Q.H., Qu, H.P.: On Hua-Tuan’s conjecture II. Sci. China Ser. A 54:1, 65–74 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhang, Q.H., Wei, J.J.: The intersection of subgroup of finite \(p\)-groups. Arch. Math. 96(1), 9–17 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. 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  MATH  Google Scholar 

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Acknowledgements

The author would like to express her sincere gratitude to the referee. Thanks to the referee for his or her valuable comments. Thanks Professor Qinhai Zhang for his frequent encouragement.

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Correspondence to Lifang Wang.

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This work was supported by NSFC (Nos. 11471198 and 11771258).

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Wang, L. Finite 2-Groups Whose Number of Subgroups of Each Order are at Most \(2^4\). Commun. Math. Stat. 6, 207–226 (2018). https://doi.org/10.1007/s40304-018-0133-1

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