Enumerating representations in finite wreath products
Let \(\{\Pi_n\}\) be either a sequence of alternating groups \(\{A_n\}\) or of symmetric groups \(\{S_n\}\). Further, let \(H\) be a finite group, \(\Gamma\) a group and \(N\) a normal subgroup of \(H\) such that \(H/N\) is Abelian. Let \[ (H\wr\Pi_n)_N:=\{(f,\pi)\in H\wr\Pi_n:\prod_if(i)\in N\},\;n\geq 0. \] The main result of the paper is computing the exponential generating function of the sequence \[ \{|\Hom(\Gamma,(H\wr\Pi_n)_N)|\}_{n=0}^\infty \] and of certain refinements of it when \((H:N)\leq 2\) and \(\Gamma\) satisfies some weak finiteness conditions. Also some special cases of the main result related to symmetric and alternating wreath products and the Weyl group of type \(D_n\) are considered. Finally, using some earlier results of the author [Combinatorica 17, No. 4, 523-554 (1997; Zbl 0906.05002)], a complete asymptotic expansion of the function \(\{|\Hom(G,H\wr S_n)|\}\) for arbitrary finite groups \(G\) and \(H\) is obtained. Among others, these results generalize those of \textit{N. Chigira} [J. Algebra 180, No. 3, 653-661 (1996; Zbl 0854.05006)] concerning the solutions of the equation \(x^n=1\) in the groups \(H\wr A_n\), \(H\wr S_n\) and \((H\wr\Pi_n)_N=W_n\).
scientific article; zbMATH DE number 1491733