Zeta functions of finite groups by enumerating subgroups

    Research output: Contribution to journalArticle

    Abstract

    For a finite group G, we consider the zeta function (Formula presented.), where H runs over the subgroups of G. First we give simple examples of abelian p-group G and non-abelian p-group G of order pm, m≥3 for odd p (resp. 2m, m≥4) for which (Formula presented.). Hence we see there are many non-abelian groups whose zeta functions have symmetry and Euler product, like the case of abelian groups. On the other hand, we show that ζG(s) determines the isomorphism class of G within abelian groups, by estimating the number of subgroups of abelian p-groups. Finally we study the problem which abelian p-group is associated with a non-abelian group having the same zeta function.

    Original languageEnglish
    Pages (from-to)1-12
    Number of pages12
    JournalCommunications in Algebra
    DOIs
    Publication statusAccepted/In press - 2017 Jan 7

    Fingerprint

    P-groups
    Riemann zeta function
    Finite Group
    Subgroup
    Abelian group
    Euler Product
    Isomorphism Classes
    p.m.
    Odd
    Symmetry

    Keywords

    • Enumerating subgroups of abelian p-groups
    • enumerating subgroups of finite groups
    • local densities of square matrices
    • zeta functions of finite groups

    ASJC Scopus subject areas

    • Algebra and Number Theory

    Cite this

    Zeta functions of finite groups by enumerating subgroups. / Hironaka, Yumiko.

    In: Communications in Algebra, 07.01.2017, p. 1-12.

    Research output: Contribution to journalArticle

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