Normal zeta function of a group: Difference between revisions

Latest revision as of 23:14, 16 August 2013

Let $G$ be a group. The normal zeta function of $G$ is defined as:

$\zeta _{G}(s)=\sum _{n=1}^{\infty }a_{n}(G)n^{-s}$

where $a_{n}(G)$ denotes the number of normal subgroups of $G$ of index $n$ . Equivalently, it is:

$\sum _{H{\underline {\triangleleft }}_{f}G}[G:H]^{-s}$

Subgroups of finite index in nilpotent groups by F. J. Grunewald, D. Segal and G. C. Smith, Inventiones mathematicae, Volume 93,Number 1, Page 185 - 223(Year 1988): ^{Gated copy (PDF)}^{More info}

@@ Line 5: / Line 5: @@
 <math>\zeta_G(s) = \sum_{n=1}^{\infty} a_n(G)n^{-s}</math>
-where <math>a_n(G)</math> denotes the number of normal subgroups of <math>G</math> of index <math>n</math>. Equivalently, it is:
+where <math>a_n(G)</math> denotes the number of [[normal subgroup]]s of <math>G</math> of [[index of a subgroup|index]] <math>n</math>. Equivalently, it is:
 <math>\sum_{H \underline{\triangleleft}_f G} [G:H]^{-s}</math>
-summing up over all [[defining ingredient::nomrla subgroup of finite index|normal subgroups of finite index]] in <math>G</math>.
+summing up over all [[defining ingredient::normal subgroup of finite index|normal subgroups of finite index]] in <math>G</math>.
 The coefficients <math>a_n(G)</math> are all finite when the group <math>G</math> is [[finitely generated group|finitely generated]]. This follows from [[finitely generated implies finitely many homomorphisms to any finite group]] (see also [[group with finitely many homomorphisms to any finite group]]).
@@ Line 16: / Line 16: @@
 * [[Zeta function of a group]]
+==References==
+===Journal references===
+* {{paperlink|GrunewaldSegalSmith}}