Treffer: Some theorems on wreath products

Let H be a subgroup of index n in a group G. The embedding theorem constructs some standard embeddings \(\phi\) of G in the unrestricted permutational wreath product \(W=H Wr S_ n\). For each homomorphism \(\alpha\) : \(H\to A\) there exists a homomorphism \(\alpha\) Wr \(S_ n: W\to A Wr S_ n\). Let \(\alpha\) \(\uparrow\) denote the composition of \(\phi\) and \(\alpha\) Wr \(S_ n\). Theorem 1 gives necessary and sufficient conditions under which a homomorphism \(G\to W\) is one of the embeddings given by the embedding theorem. Theorem \(1'\), as a version of theorem 1, gives necessary and sufficient conditions under which for any homomorphism \(\gamma\) : \(G\to A Wr S_ I\) (I denotes a fixed set) there is a subgroup H of index \(| I|\) in G and a homomorphism \(\alpha\) : \(H\to A\) such that \(\alpha \uparrow =\gamma\). In theorem 2 the author proves that \(C_ W(G\phi)\cong C_ G(H)\) for any standard embedding \(\phi\) and, if G is finite, the number of distinct such \(\phi\) is therefore \((n-1)!| H|^{n-1}| G:C_ G(H)|\). Theorem \(2'\) is an analogy of theorem 2 for \(\alpha\) \(\uparrow\). The theorems proved depend on preceding results of the author [Arch. Math. 45, 111-115 (1985; Zbl 0575.20028); 47, 309-311 (1986; Zbl 0604.20031)].