### 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 p^{m}, m≥3 for odd p (resp. 2^{m}, 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 ζ_{G}(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 language | English |
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Pages (from-to) | 1-12 |

Number of pages | 12 |

Journal | Communications in Algebra |

DOIs | |

Publication status | Accepted/In press - 2017 Jan 7 |

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### 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.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Zeta functions of finite groups by enumerating subgroups

AU - Hironaka, Yumiko

PY - 2017/1/7

Y1 - 2017/1/7

N2 - 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.

AB - 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.

KW - Enumerating subgroups of abelian p-groups

KW - enumerating subgroups of finite groups

KW - local densities of square matrices

KW - zeta functions of finite groups

UR - http://www.scopus.com/inward/record.url?scp=85008675580&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85008675580&partnerID=8YFLogxK

U2 - 10.1080/00927872.2016.1236929

DO - 10.1080/00927872.2016.1236929

M3 - Article

AN - SCOPUS:85008675580

SP - 1

EP - 12

JO - Communications in Algebra

JF - Communications in Algebra

SN - 0092-7872

ER -