Zeta functions of finite groups by enumerating subgroups

2014/10/16 by Yumiko Hironaka, Hironaka, Yumiko
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Graph theory and applications #math.GR #math.NT #msc:11M41 #msc:20E07 #msc:20K01

paper · pdf · doi:10.48550/arxiv.1410.4326

16 pages To corrected some typos and enlarge the final remark

arxiv created 2015/12/10 · arxiv updated 2015/12/11

Abstract

For a finite group G, we consider the zeta function ζ_G(s) = ∑_H \absH^-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) = ζ_G'(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.

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