Enumerating representations in finite wreath products

Let Gamma be a group (finite or infinite), H a finite group, and let R-n denote the sequence H (sic) S-n of symmetric wreath products as well as certain variants of it (including in particular H (sic) A(n) and W-n, the Weyl group of type D-n). We compute the exponential generating function for the number \Hom(Gamma, R-n)\ of Gamma-representations in R-n and for some refinements of this sequence under very mild finiteness assumptions on Gamma (always met for instance if Gamma is finitely generated). This generalizes in a uniform way the connection between the problem of counting finite index subgroups in a group rand the enumeration of Gamma-actions on finite sets on the one hand, and the recent results of Chigira concerning solutions of the equation x(m) = 1 in the groups H (sic) S-n, H (sic) A(n), and W-n on the other. We also study the asymptotics of the function \Hom(G, H (sic) S-n)\ for arbitrary finite groups G and H. (C) 2000 Academic Press.