Two results on wreath products (Q1064405)

The central fact about unrestricted permutational wreath products is the embedding theorem. This theorem asserts the existence of an embedding of a group G into the unrestricted permutational wreath product \(W=A Wr G\sigma\) where \(A\leq G\) and \(\sigma\) : \(G\to Sym I\) is a permutation representation of G on the set of right cosets of A in G. The first result of this paper (the uniqueness theorem) gives necessary and sufficient conditions for the conjugacy of two homomorphisms of a group into W by an inner automorphism of W induced by an element of the base group \(A^ I\). The uniqueness theorem leads to the second result of the paper (the centralizer theorem) which describes the centralizer of a subgroup in A Wr (Sym I). The paper closes by an application of the centralizer theorem to primitive permutation groups.