Two results on wreath products

Among the familiar basic facts concerning unrestricted permutational wreath products (sketched in the first few paragraphs of the last section of Cossey, Kegel, and Kov~cs [2]), the Embedding Theorem has the central role. It asserts the existence of an embedding; the proof is a construction which involves arbitrarily chosen coset representatives, with the effect that the embedding constructed may vary by an inner automorphism induced by an element of the base group. The theorem becomes even more useful once we rule out the possibility that a totally different construction could yield "equally natural" embeddings beyond this range of variation. The first result to be presented here achieves this in a somewhat more general context. Its essence first appeared as a special case of the last part of Theorem 3.1 in Gross and Kov/tcs [3] (re-stated as Theorem 3.02 in [4] and extensively used there as well). A full statement is given here and proved from first principles. It leads to the second result which concerns centralizers of certain subgroups in wreath products and has an application in the context of primitive permutation groups.