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EMBEDDING PERMUTATION GROUPS INTO WREATH PRODUCTS IN PRODUCT ACTION

Part of: Representation theory of groups Permutation 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: [email protected])
CSABA SCHNEIDER
Affiliation:
Centro de Álgebra da Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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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.

References

[1]Aschbacher, M., ‘Overgroups of primitive groups’, J. Aust. Math. Soc. 87 (2009), 37–82.CrossRefGoogle Scholar
[2]Aschbacher, M., ‘Overgroups of primitive groups. II’, J. Algebra 322 (2009), 1586–1626.CrossRefGoogle Scholar
[3]Aschbacher, M. and Shareshian, J., ‘Restrictions on the structure of subgroup lattices of finite alternating and symmetric groups’, J. Algebra 322 (2009), 2449–2463.CrossRefGoogle Scholar
[4]Baddeley, R. W., Praeger, C. E. and Schneider, C., ‘Transitive simple subgroups of wreath products in product action’, J. Aust. Math. Soc. 77 (2004), 55–72.CrossRefGoogle Scholar
[5]Baddeley, R. W., Praeger, C. E. and Schneider, C., ‘Innately transitive subgroups of wreath products in product action’, Trans. Amer. Math. Soc. 358 (2006), 1619–1641.CrossRefGoogle Scholar
[6]Baddeley, R. W., Praeger, C. E. and Schneider, C., ‘Intransitive Cartesian decompositions preserved by innately transitive permutation groups’, Trans. Amer. Math. Soc. 360 (2008), 743–764.CrossRefGoogle Scholar
[7]Bhattacharjee, M., Macpherson, D., Möller, R. G. and Neumann, P. M., Notes on Infinite Permutation Groups, Texts and Readings in Mathematics, 12 (Hindustan Book Agency, New Delhi, 1997; co-published by Springer, Berlin, 1997).CrossRefGoogle Scholar
[8]Cameron, P. J., Permutation Groups, London Mathematical Society Student Texts, 45 (Cambridge University Press, Cambridge, 1999).CrossRefGoogle Scholar
[9]Dixon, J. D. and Mortimer, B., Permutation Groups, Graduate Texts in Mathematics, 163 (Springer, New York, 1996).CrossRefGoogle Scholar
[10]Gillespie, N., ‘Neighbour transitive codes’, PhD Thesis, The University of Western Australia, 2011.Google Scholar
[11]Giudici, M. and Praeger, C. E., ‘Completely transitive codes in Hamming graphs’, European J. Combin. 20 (1999), 647–661.CrossRefGoogle Scholar
[12]Kovács, L. G., ‘Wreath decompositions of finite permutation groups’, Bull. Aust. Math. Soc. 40 (1989), 255–279.CrossRefGoogle Scholar
[13]Praeger, C. E., ‘An O’Nan–Scott theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs’, J. Lond. Math. Soc. (2) 47 (1993), 227–239.CrossRefGoogle Scholar
[14]Praeger, C. E. and Schneider, C., ‘Three types of inclusions of innately transitive permutation groups into wreath products in product action’, Israel J. Math. 158 (2007), 65–104.CrossRefGoogle Scholar