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EMBEDDING PERMUTATION GROUPS INTO WREATH PRODUCTS IN PRODUCT ACTION

Part of: Permutation groups Representation theory of groups Graph theory

Published online by Cambridge University Press:  24 April 2012

CHERYL E. PRAEGER  and
CSABA SCHNEIDER
CHERYL E. PRAEGER*
Affiliation:
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: cheryl.praeger@uwa.edu.au)
CSABA SCHNEIDER
Affiliation:
Centro de Álgebra da Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal (email: csaba.schneider@gmail.com)
*
✉For correspondence; e-mail: cheryl.praeger@uwa.edu.au
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Abstract

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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.

Keywords

wreath productsproduct actionpermutation groupsembedding theorems

MSC classification

Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures 20B35: Subgroups of symmetric groups 20D99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 92 , Issue 1 , February 2012 , pp. 127 - 136
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

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.

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