EMBEDDING PERMUTATION GROUPS INTO WREATH PRODUCTS IN PRODUCT ACTION

We consider the wreath product of two permutation groups G <= Sym and H <= Sym Delta as a permutation group acting on the set Pi of functions from Delta to Gamma. Such groups play an important role in the O'Nan-Scott theory of permutation groups and they also arise as automorphism groups of graph products and codes. Let X be a subgroup of Sym Gamma (sic) Sym Delta. Our main result is that, in a suitable conjugate of X, the subgroup of Sym Gamma induced by a stabiliser of a coordinate delta is an element of Delta only depends on the orbit of delta under the induced action of X on Delta. Hence, if X is transitive on Delta, then X can be embedded into the wreath product of the permutation group induced by the stabiliser X-delta on Gamma and the permutation group induced by X on Delta. We use this result to describe the case where X is intransitive on Delta and offer an application to error-correcting codes in Hamming graphs.