P. Hall's strange formula for abelian \(p\)-groups
type: Item
instance of: scholarly article
title: P. Hall's strange formula for abelian \(p\)-groups
zbMATH Open document ID: 0789.20023
author: Tomoyuki Yoshida
published in: Osaka Journal of Mathematics
publication date: 1993-06-29T00:00:00Z
review text: Let \(G\) denote a finite group and \(A\) an abelian \(p\)-group; we denote by \(s(A,G)\) the number of subgroups of \(G\) isomorphic to \(A\) and by \(h(A,G)\) the number of homomorphisms from \(A\) to \(G\). Moreover we define the zeta functions of Sylow and Frobenius types by \[ S_ G^ Ap(z) =\sum_ A{}' s(A,G)| A|^{-z}, \quad H_ G^ Ap(z) = \sum_ A{}^ \prime {h(A,G)\over|\text{Aut}(A)|}| A|^{-z} \] where \(A\) runs over a complete set of representatives of isomorphism classes of abelian \(p\)-groups. The main result is the following theorem: For any finite group \(G\) and \(\text{Re }z > -1\) \[ {H_ G^ Hp(z)\over S_ G^ Ap(z)} = \prod^ \infty_{m = 1} (1 - p^{-m-z})^{-1}. \] In particular, the left hand side is independent of the finite group \(G\). As special cases, the author gets P. Hall's strange formula \(\sum_ A'1/| \text{Aut}(A)| = \sum_ A'1/| A|\) and the fundamental theorem of finite abelian groups. The names for the zeta series above are justified because they appeared in the study of Sylow's third theorem and Frobenius' theorem on the number of solutions of the equation \(x^ n = 1\) on a finite group.
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20K01
Mathematics Subject Classification ID: 11N45
zbMATH DE Number: 219883
zbMATH Keywords: finite group
zbMATH Keywords: abelian \(p\)-group
zbMATH Keywords: number of subgroups
zbMATH Keywords: number of homomorphisms
zbMATH Keywords: zeta functions
zbMATH Keywords: fundamental theorem of finite abelian groups
zbMATH Keywords: zeta series
reviewed by: Grigore Călugăreanu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 219883