Finite ๐‘-groups all of whose ๐“2-subgroups are generated by two elements

Lihua Zhang 1 ย and Junqiang Zhang 2
  • 1 School of Sciences, Beijing University of Posts and Telecommunications, 100876, Beijing, P. R. China
  • 2 Mathematics Department, Shanxi Normal University, Shanxi 041004, Linfen, P. R. China
Lihua Zhang and Junqiang Zhang
  • Corresponding author
  • Mathematics Department, Shanxi Normal University, Linfen, Shanxi 041004, P. R. China
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar

Abstract

Assume ๐บ is a finite ๐‘-group. We prove that if all A2-subgroups of ๐บ are generated by two elements, then so are all non-abelian subgroups of ๐บ. By using this result, we classify the ๐‘-groups which have at least two A1-subgroups and in which the intersection of every pair of distinct A1-subgroups equals the intersection of all the A1-subgroups. It turns out that such ๐‘-groups are the finite ๐‘-groups all of whose non-abelian subgroups are generated by two elements, and the converse also almost holds.

  • [1]

    Y. Berkovich, Groups of Prime Power Order. Vol. 1, Walter de Gruyter, Berlin, 2008.

  • [2]

    Y. Berkovich and Z. Janko, Groups of prime power order. Vol. 2, Walter de Gruyter, Berlin, 2008.

  • [3]

    Y. Berkovich and Z. Janko, Structure of finite ๐‘-groups with given subgroups, Ischia Group Theory 2004, Contemp. Math. 402, American Mathematical Society, Providence (2006), 13โ€“93.

  • [4]

    Y. Berkovich and Q. Zhang, On A 1 \mathscr{A}_{1}- and A 2 \mathscr{A}_{2}-subgroups of finite ๐‘-groups, J. Algebra Appl. 13 (2014), no. 2, Article ID 1350095.

  • [5]

    N. Blackburn, On a special class of ๐‘-groups, Acta Math. 100 (1958), 45โ€“92.

    • Crossref
    • Export Citation
  • [6]

    N. Blackburn, Generalizations of certain elementary theorems on ๐‘-groups, Proc. Lond. Math. Soc. (3) 11 (1961), 1โ€“22.

  • [7]

    B. Huppert, Endliche Gruppen. I, Springer, Berlin, 1967.

  • [8]

    P. Li and Q. Zhang, Finite ๐‘-groups all of whose subgroups of class 2 are generated by two elements, J. Korean Math. Soc. 56 (2019), no. 3, 739โ€“750.

  • [9]

    G.โ€‰A. Miller, Number of abelian subgroups in every prime power group, Amer. J. Math. 51 (1929), no. 1, 31โ€“34.

    • Crossref
    • Export Citation
  • [10]

    M. Newman and M. Xu, Metacyclic groups of prime-power order (Research announcement), Adv. Math. (China) 17 (1988), 106โ€“107.

  • [11]

    L. Rรฉdei, Das โ€œschiefe Produktโ€ in der Gruppentheorie mit Anwendung auf die endlichen nichtkommutativen Gruppen mit lauter kommutativen echten Untergruppen und die Ordnungszahlen, zu denen nur kommutative Gruppen gehรถren, Comment. Math. Helv. 20 (1947), 225โ€“264.

  • [12]

    M. Xu, L. An and Q. Zhang, Finite ๐‘-groups all of whose non-abelian proper subgroups are generated by two elements, J. Algebra 319 (2008), no. 9, 3603โ€“3620.

    • Crossref
    • Export Citation
  • [13]

    M. Xu and Q. Zhang, A classification of metacyclic 2-groups, Algebra Colloq. 13 (2006), no. 1, 25โ€“34.

    • Crossref
    • Export Citation
  • [14]

    J. Zhang and X. Li, Finite ๐‘-groups all of whose proper subgroups have small derived subgroups, Sci. China Math. 53 (2010), no. 5, 1357โ€“1362.

    • Crossref
    • Export Citation
  • [15]

    L. Zhang, The intersection of nonabelian subgroups of finite ๐‘-groups, J. Algebra Appl. 16 (2017), no. 1, Article ID 1750020.

  • [16]

    Q. Zhang, X. Sun, L. An and M. Xu, Finite ๐‘-groups all of whose subgroups of index p 2 p^{2} are abelian, Algebra Colloq. 15 (2008), no. 1, 167โ€“180.

    • Crossref
    • Export Citation
  • [17]

    Q. Zhang, L. Zhao, M. Li and Y. Shen, Finite ๐‘-groups all of whose subgroups of index p 3 p^{3} are abelian, Commun. Math. Stat. 3 (2015), no. 1, 69โ€“162.

    • Crossref
    • Export Citation
Purchase article
Get instant unlimited access to the article.
$42.00
Log in
Already have access? Please log in.


or
Log in with your institution

Journal + Issues

The Journal of Group Theory is devoted to the publication of original research articles in all aspects of group theory. Articles concerning applications of group theory and articles from research areas which have a significant impact on group theory will also be considered.

Search