Embedding permutation groups into wreath products in product action

Abstract

We consider the wreath product of two permutation groups G ≤ Sym Γ and H ≤ Sym Δ as a permutation group acting on the set Π of functions from Δ to Γ. 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 Γ Sym Δ. Our main result is that, in a suitable conjugate of X, the subgroup of Sym Γ induced by a stabiliser of a coordinate δ ∈ Δ only depends on the orbit of δ under the induced action of X on Δ. Hence, if X is transitive on Δ, then X can be embedded into the wreath product of the permutation group induced by the stabiliser X δ on Δ and the permutation group induced by X on Δ. We use this result to describe the case where X is intransitive on Δ and offer an application to error-correcting codes in Hamming graphs. © 2012 Australian Mathematical Publishing Association Inc.