A generating function for the number of homomorphisms from a finitely generated abelian group to an alternating group
Let \(h(A,G)\) be the number of homomorphisms from a finitely generated group \(A\) to a finite group \(G\). Let \(\varepsilon\) denote the identity of a group. For a cyclic group \(C_d\) of order \(d\), \(h(C_d,G)\) coincides with the number of solutions of \(X^d=\varepsilon\) in \(G\). Let \(S_n\) be the symmetric group on \(n\) letters, and let \(A_n\) be the alternating group on \(n\) letters. For each finite group \(G\), \(G\wr S_n\) and \(G\wr A_n\) denote wreath products. \textit{N. Chigira} [J. Algebra 180, 653-661 (1996; Zbl 0854.05006)] found the exponential generating functions of \(h(C_d,G\wr S_n)\), \(h(C_d,G\wr A_n)\), \(h(C_d,W(D_n))\), where \(W(D_n)\) is the Weyl group. Let \(\Phi_2(A)\) denote the intersection of all maximal subgroups of index 2 in \(A\), and let \({\mathcal K}_A\) denote the set of subgroups \(D\) of \(A\) containing \(\Phi_2(A)\) as a subgroup of index 2. The author generalizes the above-mentioned result by deriving a generating function of \(h(A,G\wr A_n)\) in the case where \(A\) is a finitely generated abelian group.
scientific article; zbMATH DE number 1749641