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Bounds for finite semiprimitive permutation groups: order, base size, and minimal degree

Part of: Permutation groups

Published online by Cambridge University Press:  05 November 2020

Luke Morgan ,
Cheryl E. Praeger  and
Kyle Rosa
Luke Morgan
Affiliation:
University of Primorska, UP IAM, Muzejski trg 2, Koper6000, Slovenia University of Primorska, UP FAMNIT, Glagoljaška 8, Koper6000, Slovenia (luke.morgan@famnit.upr.si)
Cheryl E. Praeger
Affiliation:
Department of Mathematics and Statistics, Centre for the Mathematics of Symmetry and Computation, The University of Western Australia, 35 Stirling Highway, Crawley, WA6009, Australia (cheryl.praeger@uwa.edu.au; kylerosa@gmail.com)
Kyle Rosa
Affiliation:
Department of Mathematics and Statistics, Centre for the Mathematics of Symmetry and Computation, The University of Western Australia, 35 Stirling Highway, Crawley, WA6009, Australia (cheryl.praeger@uwa.edu.au; kylerosa@gmail.com)
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Abstract

In this paper, we study finite semiprimitive permutation groups, that is, groups in which each normal subgroup is transitive or semiregular. These groups have recently been investigated in terms of their abstract structure, in a similar way to the O'Nan–Scott Theorem for primitive groups. Our goal here is to explore aspects of such groups which may be useful in place of precise structural information. We give bounds on the order, base size, minimal degree, fixed point ratio, and chief length of an arbitrary finite semiprimitive group in terms of its degree. To establish these bounds, we study the structure of a finite semiprimitive group that induces the alternating or symmetric group on the set of orbits of an intransitive minimal normal subgroup.

Keywords

Permutation groupsbase sizeminimal degree

MSC classification

Primary: 20B05: General theory for finite groups
Secondary: 20B15: Primitive groups
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 4 , November 2020 , pp. 1071 - 1091
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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References

Babai, L., On the order of uniprimitive permutation groups, Ann. Math. (2) 113(3) (1981), 553–568.CrossRefGoogle Scholar
Babai, L., On the order of doubly transitive permutation groups, Invent. Math. 65 (1982), 473–484.CrossRefGoogle Scholar
Bamberg, J., Bounds and quotient actions of innately transitive groups, J. Aust. Math. Soc. 79(1) (2005), 95–112.CrossRefGoogle Scholar
Bamberg, J. and Praeger, C. E., Finite permutation groups with a transitive minimal normal subgroup, Proc. London Math. Soc. 89(3) (2004), 71–103.CrossRefGoogle Scholar
Bereczky, Á. and Maróti, A., On groups with every normal subgroup transitive or semiregular, J. Algebra 319(4) (2008), 1733–1751.CrossRefGoogle Scholar
Bosma, W., Cannon, J. and Playoust, C., The Magma algebra system. I. The user language, J. Symbolic Comput. 24(3–4) (1997), 235–265. Computational algebra and number theory (London, 1993).CrossRefGoogle Scholar
Burness, T. C., Simple groups, fixed point ratios and applications, in Local representation theory and simple groups, pp. 267–322, EMS Ser. Lect. Math. (Eur. Math. Soc., Zürich, 2018).CrossRefGoogle Scholar
Detomi, E. and Lucchini, A., Invariable generation of permutation groups, Arch. Math. (Basel) 104(4) (2015), 301–309.CrossRefGoogle Scholar
Devillers, A., Harper, S. and Morgan, L., The distinguishing number of quasiprimitive and semiprimitive groups, Arch. Math. (Basel) 113 (2019), 127–139.CrossRefGoogle Scholar
Frohardt, D. and Magaard, K., Monodromy composition factors among exceptional groups of Lie type, pp. 134–143, Group theory (Granville, OH, 1992) (World Sci. Publ., River Edge, NJ, 1993)Google Scholar
Giudici, M. and Morgan, L., A class of semiprimitive groups that are graph-restrictive, Bull. Lond. Math. Soc. 46(6) (2014), 1226–1236.CrossRefGoogle Scholar
Giudici, M. and Morgan, L., On locally semiprimitive graphs and a theorem of Weiss, J. Algebra 427 (2015), 104–117.CrossRefGoogle Scholar
Giudici, M. and Morgan, L., A theory of semiprimitive groups, J. Algebra 503 (2018), 146–185.CrossRefGoogle Scholar
Glasby, S. P., Praeger, C. E., Rosa, K. and Verret, G., Bounding the composition length of primitive permutation groups and completely reducible linear groups, J. London Math. Soc. 98(3) (2018), 557–572.CrossRefGoogle Scholar
Guralnick, R. and Liebeck, M., Permutation representations of nonsplit extensions involving alternating groups, Israel J. Math. 229 (2019), 181–191.CrossRefGoogle Scholar
Guralnick, R. and Magaard, K., On the minimal degree of a primitive permutation group, J. Algebra 207(1) (1998), 127–145.CrossRefGoogle Scholar
Guralnick, R. M., Maróti, A. and Pyber, L., Normalizers of primitive permutation groups, Adv. Math. 310 (2017), 1017–1063.CrossRefGoogle Scholar
Jordan, C., Sur la limite de transitivité des groupes non alternés, Bull. Soc. Math. France 1 (1872/73), 40–71.CrossRefGoogle Scholar
Jordan, C., Sur la limite de degré des groupes primitifs qui contiennent une substitution donnée, J. Reine Angew. Math. 79 (1875), 248–258.Google Scholar
Kempe, J., Pyber, L. and Shalev, A., Permutation groups, minimal degrees and quantum computing, Groups Geom. Dyn. 1(4) (2007), 553–584.CrossRefGoogle Scholar
Liebeck, M. W. and Saxl, J., Minimal degrees of primitive permutation groups, with an application to monodromy groups of covers of Riemann surfaces, Proc. London Math. Soc. Third Ser. 63(2) (1991), 266–314.CrossRefGoogle Scholar
Lucchini, A., Menegazzo, F. and Morigi, M., On the number of generators and composition length of finite linear groups, J. Algebra 243(2) (2001), 427–447.CrossRefGoogle Scholar
Menezes, N. E., Random generation and chief length of finite groups, PhD Thesis, University of St Andrews (2013)Google Scholar
Potočnik, P., Spiga, P. and Verret, G., On the order of arc-stabilisers in arc-transitive graphs with prescribed local group, Trans. Am. Math. Soc. 366(7) (2014), 3729–3745.CrossRefGoogle Scholar
Praeger, C. E., An O'Nan-Scott theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs, J. London Math. Soc. (2) 47(2) (1993), 227–239.CrossRefGoogle Scholar
Praeger, C. E., Seminormal and subnormal subgroup lattices for transitive permutation groups, J. Aust. Math. Soc. 80 (2006), 45–63.CrossRefGoogle Scholar
Praeger, C. E. and Saxl, J., On the orders of primitive permutation groups, Bull. Lond. Math. Soc. 12(4) (1980), 303–307.CrossRefGoogle Scholar
Praeger, C. E. and Schneider, C., Permutation groups and cartesian decompositions, London Mathematical Society Lecture Note Series, Volume 449 (Cambridge University Press, 2018).CrossRefGoogle Scholar
Praeger, C. E. and Shalev, A., Bounds on finite quasiprimitive permutation groups, J. Australian Math. Soc. 71(2) (2001), 243–258. Special issue on group theory.CrossRefGoogle Scholar
Praeger, C. E., Pyber, L., Spiga, P. and Szabó, E., Graphs with automorphism groups admitting composition factors of bounded rank, Proc. Am. Math. Soc. 140(7) (2012), 2307–2318.CrossRefGoogle Scholar
Pyber, L., Asymptotic results for permutation groups, pp. 197–219, Groups and Computation 1993 (New Brunswick, NJ, 1991)CrossRefGoogle Scholar
Rosa, K., Structure, covers, and bounds of semiprimitive groups, Masters Thesis, The University of Australia (2018)Google Scholar
Rotman, J. J., A first course in abstract algebra (Upper Saddle River, NJ, Prentice Hall, Inc., 1996).Google Scholar
Sims, C. C., Computational methods in the study of permutation groups, Computational problems in abstract algebra, pp. 169–183 (Pergamon Press, Oxford, 1970).Google Scholar
Spiga, P. and Verret, G., On intransitive graph-restrictive permutation groups, J. Algebraic Combin. 40(1) (2014), 179–185.CrossRefGoogle Scholar
Trofimov, V. I. and Weiss, R. M., Graphs with a locally linear group of automorphisms, Math. Proc. Cambridge Philos. Soc. 118(2) (1995), 191–206.CrossRefGoogle Scholar
Wagner, A., The faithful linear representation of least degree of S n and A n over a field of characteristic 2, Mathematische Zeitschrift 151(2) (1976), 127–137.CrossRefGoogle Scholar
Wagner, A., The faithful linear representations of least degree of S n and A n over a field of odd characteristic, Mathematische Zeitschrift 154(2) (1977), 103–114.CrossRefGoogle Scholar
Weiss, R., s-transitive graphs, in Algebraic methods in graph theory, Vol. I, II (Szeged, 1978), pp. 827–847, Colloq. Math. Soc. János Bolyai, Volume 23 (North-Holland, Amsterdam/New York, 1981)Google Scholar
Wielandt, H., Finite permutation groups (Academic Press, New York-London, 1964).Google Scholar
Wielandt, H., Permutation groups through invariant relations and invariant functions, Lecture Notes (Ohio State University, Columbus, 1969).Google Scholar
Wielandt, H., Mathematische Werke/Mathematical works, Volume 1 (Walter de Gruyter & Co., Berlin, 1994). Group theory, With essays on some of Wielandt's works by G. Betsch, B. Hartley, I. M. Isaacs, O. H. Kegel and P. M. Neumann, Edited and with a preface by Bertram Huppert and Hans Schneider.Google Scholar
Wilson, R. A., The finite simple groups, Graduate Texts in Mathematics, Volume 251 (London, Springer-Verlag London, Ltd., 2009).CrossRefGoogle Scholar