## Abstract

For a finite group G, we consider the zeta function ζ_{G}(s) = ∑_{H} ǀHǀ^{-s}, 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 ζ_{G}(s) = ζG'(s). 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) | 3365-3376 |

Number of pages | 12 |

Journal | Communications in Algebra |

Volume | 45 |

Issue number | 8 |

DOIs | |

Publication status | Published - 2017 Aug 3 |

## 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