Wreath product of permutation groups and their actions on a sets

Abstract. 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)}^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 = fs, where fs (δ) =(f. s^−1) (δ) =(f. s^−1)(δ) for all δєδ. finaly, we give the wreath product w of s (γ) by s (δ), and the action of w on γ*δ.