Innately transitive subgroups of wreath products in product action

Authors:
Robert W. Baddeley, Cheryl E. Praeger and Csaba Schneider

Journal:
Trans. Amer. Math. Soc. **358** (2006), 1619-1641

MSC (2000):
Primary 20B05, 20B15, 20B25, 20B35

DOI:
https://doi.org/10.1090/S0002-9947-05-03750-5

Published electronically:
June 21, 2005

MathSciNet review:
2186989

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Abstract | References | Similar Articles | Additional Information

Abstract: A permutation group is innately transitive if it has a transitive minimal normal subgroup, which is referred to as a plinth. We study the class of finite, innately transitive permutation groups that can be embedded into wreath products in product action. This investigation is carried out by observing that such a wreath product preserves a natural Cartesian decomposition of the underlying set. Previously we classified the possible embeddings in the case where the plinth is simple. Here we extend that classification and identify several different types of Cartesian decompositions that can be preserved by an innately transitive group with a non-abelian plinth. These different types of decompositions lead to different types of embeddings of the acting group into wreath products in product action. We also obtain a full characterisation of embeddings of innately transitive groups with diagonal type into such wreath products.

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Additional Information

**Robert W. Baddeley**

Affiliation:
32 Arbury Road, Cambridge CB4 2JE, United Kingdom

Email:
robert.baddeley@ntworld.com

**Cheryl E. Praeger**

Affiliation:
School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway 6009 Crawley, Western Australia

Email:
praeger@maths.uwa.edu.au

**Csaba Schneider**

Affiliation:
Informatics Laboratory, Computer and Automation Research Institute of the Hungarian Academy of Sciences, P.O. Box 63, 1518 Budapest, Hungary

Email:
csaba.schneider@sztaki.hu

DOI:
https://doi.org/10.1090/S0002-9947-05-03750-5

Keywords:
Innately transitive groups,
plinth,
characteristically simple groups,
Cartesian decompositions,
Cartesian systems

Received by editor(s):
December 18, 2003

Received by editor(s) in revised form:
May 28, 2004

Published electronically:
June 21, 2005

Additional Notes:
The authors acknowledge the support of an Australian Research Council grant. The third author was employed by The University of Western Australia as an ARC Research Associate while the research presented in this paper was carried out. We are very grateful to Laci Kovács for explaining the origins of some of the ideas that appear in this article.

Article copyright:
© Copyright 2005
American Mathematical Society