P. Hall's strange formula for abelian \(p\)-groups (Q1802978)

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.