Innately transitive subgroups of wreath products in product action
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- by Robert W. Baddeley, Cheryl E. Praeger and Csaba Schneider PDF
- Trans. Amer. Math. Soc. 358 (2006), 1619-1641 Request permission
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
- Robert W. Baddeley and Cheryl E. Praeger, On classifying all full factorisations and multiple-factorisations of the finite almost simple groups, J. Algebra 204 (1998), no. 1, 129–187. MR 1623953, DOI 10.1006/jabr.1997.7275
- Robert W. Baddeley and Cheryl E. Praeger, On primitive overgroups of quasiprimitive permutation groups, J. Algebra 263 (2003), no. 2, 294–344. MR 1978653, DOI 10.1016/S0021-8693(03)00113-3
- Robert W. Baddeley, Cheryl E. Praeger, and Csaba Schneider, Transitive simple subgroups of wreath products in product action, J. Aust. Math. Soc. 77 (2004), no. 1, 55–72. MR 2069025, DOI 10.1017/S1446788700010156
- John Bamberg and Cheryl E. Praeger, Finite permutation groups with a transitive minimal normal subgroup, Proc. London Math. Soc. (3) 89 (2004), no. 1, 71–103. MR 2063660, DOI 10.1112/S0024611503014631
- Barbara Baumeister, Factorizations of primitive permutation groups, J. Algebra 194 (1997), no. 2, 631–653. MR 1467170, DOI 10.1006/jabr.1997.7027
- John D. Dixon and Brian Mortimer, Permutation groups, Graduate Texts in Mathematics, vol. 163, Springer-Verlag, New York, 1996. MR 1409812, DOI 10.1007/978-1-4612-0731-3
- L. G. Kovács, Primitive subgroups of wreath products in product action, Proc. London Math. Soc. (3) 58 (1989), no. 2, 306–322. MR 977479, DOI 10.1112/plms/s3-58.2.306
- L. G. Kovács, Wreath decompositions of finite permutation groups, Bull. Austral. Math. Soc. 40 (1989), no. 2, 255–279. MR 1012834, DOI 10.1017/S0004972700004366
- 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 (1987), no. 2, 365–383. MR 916173, DOI 10.1016/0021-8693(87)90223-7
- 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 (1990), no. 432, iv+151. MR 1016353, DOI 10.1090/memo/0432
- Cheryl E. Praeger, The inclusion problem for finite primitive permutation groups, Proc. London Math. Soc. (3) 60 (1990), no. 1, 68–88. MR 1023805, DOI 10.1112/plms/s3-60.1.68
- Cheryl E. Praeger, An O’Nan-Scott theorem for finite quasiprimitive permutation groups and an application to $2$-arc transitive graphs, J. London Math. Soc. (2) 47 (1993), no. 2, 227–239. MR 1207945, DOI 10.1112/jlms/s2-47.2.227
- Cheryl E. Praeger and Csaba Schneider, Factorisations of characteristically simple groups, J. Algebra 255 (2002), no. 1, 198–220. MR 1935043, DOI 10.1016/S0021-8693(02)00111-4
- Cheryl E. Praeger and Csaba Schneider, Ordered triple designs and wreath products of groups, Statistics and science: a Festschrift for Terry Speed, IMS Lecture Notes Monogr. Ser., vol. 40, Inst. Math. Statist., Beachwood, OH, 2003, pp. 103–113. MR 2004334, DOI 10.1214/lnms/1215091137
- 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.
- Leonard L. Scott, Representations in characteristic $p$, The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979) Proc. Sympos. Pure Math., vol. 37, Amer. Math. Soc., Providence, R.I., 1980, pp. 319–331. MR 604599
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
- MR Author ID: 141715
- ORCID: 0000-0002-0881-7336
- 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
- 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.
- © Copyright 2005 American Mathematical Society
- 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
- MathSciNet review: 2186989