a1 Centre for Mathematics of Symmetry and Computation, School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia (email: email@example.com)
a2 Centro de Álgebra da Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal (email: firstname.lastname@example.org)
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 SymΓ 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 Xδ on Γ 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.
(Received April 28 2011)
(Accepted January 16 2012)
2010 Mathematics subject classification
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Praeger is supported by Australian Research Council Federation Fellowship FF0776186. Schneider acknowledges the support of the grants PEst-OE/MAT/UI0143/2011 and PTDC/MAT/101993/2008 of the Fundação para a Ciência e a Tecnologia (Portugal) and of the Hungarian Scientific Research Fund (OTKA) grant 72845.
Communicated by I. E. Shparlinski
This paper is dedicated to the memory of Alf van der Poorten