Zeta functions of finite groups by enumerating subgroups. Issue 8 (3rd August 2017)

• Record Type: Journal Article
• Authors: Hironaka, Yumiko
• Is Part Of: Communications in algebra. Volume 45:Issue 8(2017)
• Page Start: 3365
• Page End: 3376
• Publication Date: 2017-08-03
• Subjects: Enumerating subgroups of abelian p-groups -- enumerating subgroups of finite groups -- local densities of square matrices -- zeta functions of finite groups; 20E07 (Primary) -- 11M41 -- 20K01; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2016.1236929
• Languages: English
• ISSNs: 0092-7872

Abstract

ABSTRACT: For a finite group G, we consider the zeta functionζ G ( s ) = ∑ H | H | − 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 whichζ 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.