Intransitive Cartesian decompositions preserved by innately transitive permutation groups

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

Journal:
Trans. Amer. Math. Soc. **360** (2008), 743-764

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

Published electronically:
August 28, 2007

MathSciNet review:
2346470

Full-text PDF Free Access

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 innately transitive group projects onto a transitive subgroup of the top group. In this article we prove that the transitivity assumption we made in the previous paper was not too restrictive. Indeed, the image of the projection into the top group can only be intransitive when the finite simple group that is involved in the plinth comes from a small list. Even then, the innately transitive group can have at most three orbits on an invariant Cartesian decomposition. A consequence of this result is that if is an innately transitive subgroup of a wreath product in product action, then the natural projection of into the top group has at most two orbits.

**[BP98]**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**, 10.1006/jabr.1997.7275**[BP03]**Robert W. Baddeley and Cheryl E. Praeger,*On primitive overgroups of quasiprimitive permutation groups*, J. Algebra**263**(2003), no. 2, 294–344. MR**1978653**, 10.1016/S0021-8693(03)00113-3**[BPS04]**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**, 10.1017/S1446788700010156**[BPS06]**Robert W. Baddeley, Cheryl E. Praeger, and Csaba Schneider,*Innately transitive subgroups of wreath products in product action*, Trans. Amer. Math. Soc.**358**(2006), no. 4, 1619–1641 (electronic). MR**2186989**, 10.1090/S0002-9947-05-03750-5**[BPS07]**Robert W. Baddeley, Cheryl E. Praeger, and Csaba Schneider.

Quasiprimitive groups and blow-up decompositions.*J. Algebra*, 311(1):337-351, 2007.**[BamP04]**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**, 10.1112/S0024611503014631**[Atlas85]**J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson,*Atlas of finite groups*, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR**827219****[DM96]**John D. Dixon and Brian Mortimer,*Permutation groups*, Graduate Texts in Mathematics, vol. 163, Springer-Verlag, New York, 1996. MR**1409812****[Hup67]**B. Huppert,*Endliche Gruppen. I*, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR**0224703****[Kle87]**Peter B. Kleidman,*The maximal subgroups of the finite 8-dimensional orthogonal groups 𝑃Ω⁺₈(𝑞) and of their automorphism groups*, J. Algebra**110**(1987), no. 1, 173–242. MR**904187**, 10.1016/0021-8693(87)90042-1**[Kov89a]**L. G. Kovács,*Primitive subgroups of wreath products in product action*, Proc. London Math. Soc. (3)**58**(1989), no. 2, 306–322. MR**977479**, 10.1112/plms/s3-58.2.306**[Kov89b]**L. G. Kovács,*Wreath decompositions of finite permutation groups*, Bull. Austral. Math. Soc.**40**(1989), no. 2, 255–279. MR**1012834**, 10.1017/S0004972700004366**[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**(1990), no. 432, iv+151. MR**1016353**, 10.1090/memo/0432**[Pra90]**Cheryl E. Praeger,*The inclusion problem for finite primitive permutation groups*, Proc. London Math. Soc. (3)**60**(1990), no. 1, 68–88. MR**1023805**, 10.1112/plms/s3-60.1.68**[PS02]**Cheryl E. Praeger and Csaba Schneider,*Factorisations of characteristically simple groups*, J. Algebra**255**(2002), no. 1, 198–220. MR**1935043**, 10.1016/S0021-8693(02)00111-4**[PS03]**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**, 10.1214/lnms/1215091137**[PS07]**Cheryl E. Praeger and Csaba Schneider.

Three types of Cartesian decompositions preserved by innately transitive permutation groups.*Israel J. Math.*, 158:65-104, 2007.**[Sco80]**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**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
20B05,
20B15,
20B25,
20B35

Retrieve articles in all journals with MSC (2000): 20B05, 20B15, 20B25, 20B35

Additional Information

**Robert W. Baddeley**

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

Email:
robert.baddeley@ntworld.com

**Cheryl E. Praeger**

Affiliation:
Department 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 Research Laboratory, Computer and Automation Research Institute, 1518 Budapest, Pf. 63, Hungary

Email:
csaba.schneider@sztaki.hu

DOI:
https://doi.org/10.1090/S0002-9947-07-04223-7

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

Received by editor(s):
June 29, 2004

Received by editor(s) in revised form:
October 10, 2005

Published electronically:
August 28, 2007

Article copyright:
© Copyright 2007
American Mathematical Society