TY - JOUR
T1 - Zeta functions of finite groups by enumerating subgroups
AU - Hironaka, Yumiko
N1 - Publisher Copyright:
© 2017 Taylor & Francis.
PY - 2017/8/3
Y1 - 2017/8/3
N2 - For a finite group G, we consider the zeta function ζG(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 pm, m≥3 for odd p (resp. 2m, m≥4) for which ζG(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 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 ζG(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 pm, m≥3 for odd p (resp. 2m, m≥4) for which ζG(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 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
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U2 - 10.1080/00927872.2016.1236929
DO - 10.1080/00927872.2016.1236929
M3 - Article
AN - SCOPUS:85008675580
VL - 45
SP - 3365
EP - 3376
JO - Communications in Algebra
JF - Communications in Algebra
SN - 0092-7872
IS - 8
ER -