Wreath product of permutation groups and their actions on a sets

The object of wreath product of permutation groups is defined the actions on cartesian product of two sets. In this paper we consider S(Γ) and S(Δ) be permutation groups on Γ and Δ respectively, and S(Γ)^{Δ} be the set of all maps of Δ into the permutations group S(Γ). That is S(Γ)^{Δ}={f:Δ→S(Γ)}. S(Γ)^{Δ} is a group with respect to the multiplication defined by for all δ in Δ by (f₁f₂)(δ)=f₁(δ)f₂(δ). After that, we introduce the notion of S(Δ) actions on S(Γ)^{Δ} : S(Δ)×S(Γ)^{Δ}→S(Γ)^{Δ},(s,f)↦s⋅f=f^{s}, wheref^{s}(δ)=(f∘s⁻¹)(δ)=(fs⁻¹)(δ) for all δ∈Δ.Finaly, we give the wreath product W of S(Γ) by S(Δ), and the action of W on Γ×Δ.