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Innately transitive subgroups of wreath products in product action
Author(s):
Robert
W.
Baddeley;
Cheryl
E.
Praeger;
Csaba
Schneider
Journal:
Trans. Amer. Math. Soc.
358
(2006),
1619-1641.
MSC (2000):
Primary 20B05, 20B15, 20B25, 20B35
Posted:
June 21, 2005
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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.
References:
-
- [BadP98]
- Robert W. Baddeley and Cheryl E. Praeger.
On classifying all full factorisations and multiple-factorisations of the finite almost simple groups,
J. Algebra, 204(1):129-187, 1998. MR 1623953 (99h:20019) - [BadP03]
- R. W. Baddeley and C. E. Praeger.
On primitive overgroups of quasiprimitive permutation groups,
J. Algebra, 263(2):294-344, 2003. MR 1978653 (2004b:20005) - [BPS04]
- Robert W. Baddeley, Cheryl E. Praeger and Csaba Schneider.
Transitive simple subgroups of wreath products in product action,
J. Austral. Math. Soc., 77(1):55-72, 2004. MR 2069025 - [BamP04]
-
John Bamberg and Cheryl E. Praeger. Finite permutation groups with a transitive minimal normal subgroup,
Proc. London. Math. Soc. 89(1):71-103, 2004. MR 2063660 - [Bau97]
- Barbara Baumeister.
Factorizations of primitive permutation groups.
J. Algebra, 194(2):631-653, 1997. MR 1467170 (99i:20004) - [DM96]
- John D. Dixon and Brian Mortimer.
Permutation groups,
Springer-Verlag, New York, 1996. MR 1409812 (98m:20003) - [Kov89a]
- L. G. Kovács.
Primitive subgroups of wreath products in product action.
Proc. London Math. Soc. (3), 58(2):306-322, 1989. MR 0977479 (90a:20008) - [Kov89b]
- L. G. Kovács.
Wreath decompositions of finite permutation groups.
Bull. Austral. Math. Soc., 40(2):255-279, 1989. MR 1012834 (90j:20006) - [LPS87]
- Martin W. Liebeck, Cheryl E. Praeger, and Jan Saxl.
A classification of the maximal subgroups of the finite alternating and symmetric groups,
J. Algebra, 111(2):365-383, 1987. MR 0916173 (89b:20008) - [LPS90]
- Martin W. Liebeck, Cheryl E. Praeger, and Jan Saxl.
The maximal factorizations of the finite simple groups and their automorphism groups.
Mem. Amer. Math. Soc., 86(432):iv+151, 1990. MR 1016353 (90k:20048) - [Pra90]
- Cheryl E. Praeger.
The inclusion problem for finite primitive permutation groups,
Proc. London Math. Soc. (3), 60(1):68-88, 1990. MR 1023805 (90j:20009) - [Pra93]
- Cheryl E. Praeger.
An O'Nan-Scott theorem for finite quasiprimitive permutation groups and an application to  -arc transitive graphs.
J. London Math. Soc. (2), 47(2):227-239, 1993. MR 1207945 (94f:05068) - [PS02]
- Cheryl E. Praeger and Csaba Schneider.
Factorisations of characteristically simple groups.
J. Algebra, 255(1):198-220, 2002. MR 1935043 (2003i:20046) - [PS03]
- Cheryl E. Praeger and Csaba Schneider.
Ordered triple designs and wreath products of groups.
In Darlene R. Goldstein (Ed.) Science and Statistics: A Festschrift for Terry Speed. Institute of Mathematical Statistics, Lecture Notes - Monograph Series, volume 40, pages 103-113, 2003. MR 2004334 - [PS]
- Cheryl E. Praeger and Csaba Schneider.
Three types of inclusions of innately transitive permutation groups into wreath products in product action. Submitted arxiv.org/math.GR/0406600. - [Sco80]
- Leonard L. Scott.
Representations in characteristic  .
In The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), pages 319-331. Amer. Math. Soc., Providence, R.I., 1980. MR 0604599 (82e:20052)
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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:
10.1090/S0002-9947-05-03750-5
PII:
S 0002-9947(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
Posted:
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.
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Copyright
2005,
American Mathematical Society
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