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Enumerating Representations in Finite Wreath Products

Abstract Let Γ be a group (finite or infinite), H a finite group, and let Rn denote the sequence H≀Sn of symmetric wreath products as well as certain variants of it (including in particular H≀An and Wn, the Weyl group of type Dn). We compute the exponential generating function for the number |Hom(Γ, Rn)| of Γ-representations in Rn and for some refinements of this sequence under very mild finiteness assumptions on Γ (always met for instance if Γ is finitely generated). This generalizes in a uniform way the connection between the problem of counting finite index subgroups in a group Γ and the enumeration of Γ-actions on finite sets on the one hand, and the recent results of Chigira concerning solutions of the equation xm=1 in the groups H≀Sn, H≀An, and Wn on the other. We also study the asymptotics of the function |Hom(G, H≀Sn)| for arbitrary finite groups G and H.