Journal of the Australian Mathematical Society

Table of Contents - February 2012 - Volume 92, Issue 01  

Research Article

EMBEDDING PERMUTATION GROUPS INTO WREATH PRODUCTS IN PRODUCT ACTION

CHERYL E. PRAEGERa1 c1 and CSABA SCHNEIDERa2

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: cheryl.praeger@uwa.edu.au)

a2 Centro de Álgebra da Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal (email: csaba.schneider@gmail.com)

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 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

  • primary 05C25; secondary 20B05;
  • 20B25;
  • 20B35;
  • 20D99

Keywords and phrases

  • wreath products;
  • product action;
  • permutation groups;
  • embedding theorems

Correspondence:

c1 For correspondence; e-mail: cheryl.praeger@uwa.edu.au

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.

Communicated by I. E. Shparlinski

This paper is dedicated to the memory of Alf van der Poorten