Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-24T13:46:55.158Z Has data issue: false hasContentIssue false
  • English
  • Français

The primitive permutation groups of degree less than 1000

Published online by Cambridge University Press:  24 October 2008

John D. Dixon  and
Brian Mortimer
John D. Dixon
Affiliation:
Department of Mathematics and Statistics, Carleton University, Ottawa, Canada, KIS 5B6
Brian Mortimer
Affiliation:
Department of Mathematics and Statistics, Carleton University, Ottawa, Canada, KIS 5B6
Get access Rights & Permissions [Opens in a new window]

Extract

Our object is to describe all of the-primitive permutation groups of degree less than 1000 together with some of their significant properties. We think that such a list is of interest in illustrating in concrete form the kinds of primitive groups which arise, in suggesting conjectures about primitive groups, and in settling small exceptional cases which often occur in proofs of theorems about permutation groups. The range that we consider is large enough to allow examples of most of the types of primitive group to appear. Earlier lists (of varying completeness and accuracy) of primitive groups of degree d have been published by: C. Jordan (1872) [21] for d ≤ 17, by W. Burnside (1897) [5] for d ≤ 8, by Manning (1929) [34–38] for d ≤ 15, by C. C. Sims (1970) [45] for d ≤ 20, and by B. A. Pogorelev (1980) [42] for d ≤ 50. Unpublished lists have also been prepared by C. C. Sims for d ≤ 50 and by Mizutani[41] for d ≤ 48. Using the classification of finite simple groups which was completed in 1981 we have been able to extend the list much further. Our task has been greatly simplified by the detailed information about many finite simple groups which is available in the Atlas of Finite Groups which we will refer to as the Atlas [8].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 103 , Issue 2 , March 1988 , pp. 213 - 238
Copyright
Copyright © Cambridge Philosophical Society 1988

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Artin, E.. Geometric Algebra (Interscience, 1957).Google Scholar
[2]Aschbacher, M. and Scott, L.. Maximal subgroups of finite groups. J. Algebra 92 (1985), 44–80.CrossRefGoogle Scholar
[3]Bannai, E.. Maximal subgroups of low rank of finite symmetric and alternating groups. J. Fac. Sci. Univ. Tokyo 18 (1972), 457–486.Google Scholar
[4]Bloom, D. M.. The subgroups of PSL(3, q) for odd q. Trans. Amer. Math. Soc. 127 (1967) 150–178.Google Scholar
[5]Burnside, W.. Theory of Groups of Finite Order, 2nd ed. (Cambridge University Press, 1911), (reprinted, Dover, 1955).Google Scholar
[6]Cameron, P. J.. Finite permutation groups and finite simple groups. Bull. London Math. Soc. 13 (1981), 1–22.CrossRefGoogle Scholar
[7]Cameron, P. J., Neumann, P. M. and Teague, D. N.. On the degrees of primitive permutation groups. Math. Z. 180 (1982), 141–149.CrossRefGoogle Scholar
[8]Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A.. Atlas of Finite Groups (Clarendon Press, 1985).Google Scholar
[9]Cooperstein, B. N.. Minimal degree for a permutation representation of a classical group. Israel J. Math. 30 (1978), 213–235.CrossRefGoogle Scholar
[10]Darafshah, M. R.. Maximal subgroups of the group GL 6(2). Bull. Malaysian Math Soc. (2) 7 (1984), 49–55.Google Scholar
[11]Dickson, L. E.. Linear Groups With an Exposition of the Galois Field Theory (Leipzig, 1901) (reprinted, Dover, 1958).Google Scholar
[12]Dieudonné, J.. La géométrie des groupes classiques (Springer-Verlag, 1963).CrossRefGoogle Scholar
[13]Feit, W.. The current situation in the theory of finite simple groups. In Proc. Internat. Congress Math. (Nice, 1970), vol. 1 (Gauthier-Villars, 1971), pp. 55–93.Google Scholar
[14]Fischer, J. and McKay, J.. The nonabelian simple groups G, |G| < 106 – the maximal subgroups. Math. Comp. 32 (1978), 1293–1302.Google Scholar
[15]Harada, K. and Yamaki, H.. The irreducible subgroups of GLn(2) with n ≤ 6. C. R. Math. Rep. Acad. Sci. Canada 1 (1979), 75–78.Google Scholar
[16]Hartley, R. W.. Determination of the ternary collineation groups whose coefficients lie in the field GF(2n). Ann. of Math. 27 (1925), 140–158.CrossRefGoogle Scholar
[17]Hoffman, C. M.. Group-Theoretic Algorithms and Graph Isomorphism. Lecture Notes in Computer Science, vol. 136 (Springer-Verlag, 1982).CrossRefGoogle Scholar
[18]Huppert, B.. Endliche Gruppen, I (;Springer-Verlag, 1967).CrossRefGoogle Scholar
[19]Ivanov, A. A., Klin, M. Kh., Tsaranov, S. V. and Shpektorov, S. K.. On the problem of computing the subdegrees of transitive permutation groups. Uspekhi Mat. Nauk 38 (1983), 115–116 ( = Russian Math. Surveys 38 (1983), 123–124).Google Scholar
[20]Iyanaga, S. and Kawada, Y. (eds). Encyclopedic Dictionary of Mathematics (MIT Press, 1977).Google Scholar
[21]Jordan, C.. Traité des Substitutions et des Equations Algébriques (reprinted, Albert Blanchard, 1957).Google Scholar
[22]Kantor, W. M.. Permutation representations of the finite classical groups of small degree or rank. J. Algebra 60 (1979), 158–168.CrossRefGoogle Scholar
[23]Kantor, W. M. and Liebler, R. A.. The rank 3 permutation representations of the finite classical groups. Trans. Amer. Math. Soc. 271 (1982), 1–71.CrossRefGoogle Scholar
[24]Kondrat'ev, A. S.. Irreducible subgroups of GL(7,2). Mat. Zametki 37 (1985), 317–321 ( = Math. Notes 37 (1985), 178–181).Google Scholar
[25]Kondrat'ev, A. S.. Irreducible subgroups of GL(9,2). Mat. Zametki 39 (1986), 320–329 (= Math. Notes 39 (1986), 173–178).Google Scholar
[26]Landazuri, V. and Seitz, G.. On the minimal degrees of permutation representations of the finite Chevalley groups. J. Algebra 32 (1974), 418–443.CrossRefGoogle Scholar
[27]Liebeck, M. W.. On the orders of maximal subgroups of the finite classical groups. Proc. London Math. Soc. (3) 50 (1985), 426–446.CrossRefGoogle Scholar
[28]Liebeck, M. W.. The affine permutation groups of rank three. Proc. London Math. Soc. (to appear).Google Scholar
[29]Liebeck, M. W., Praeger, Cheryl and Saxl, J.. On the O'Nan–Scott Theorem for finite primitive permutation groups. J. Australian Math. Soc. (to appear).Google Scholar
[30]Liebeck, M. W. and Saxl, J.. Primitive permutation groups containing an element of large prime order. J. London Math. Soc. (2) 31 (1985), 237–249.CrossRefGoogle Scholar
[31]Liebeck, M. W. and Saxl, J.. Primitive permutation groups of odd degree. J. London Math. Soc. (2) 31 (1985), 237–249.CrossRefGoogle Scholar
[32]Liebeck, M. W. and Saxl, J.. The finite primitive permutation groups of rank three. Bull. London Math. Soc. 18 (1986), 165–172.CrossRefGoogle Scholar
[33]Luneburg, H.. Translation Planes (Springer-Verlag, 1980).CrossRefGoogle Scholar
[34]Manning, W. A.. On the primitive groups of class ten. Amer. J. Math. 28 (1906), 227–236.CrossRefGoogle Scholar
[35]Manning, W. A.. On the primitive groups of class six and eight. Amer. J. Math. 32 (1910), 235–256.CrossRefGoogle Scholar
[36]Manning, W. A.. On the primitive groups of class twelve. Amer. J. Math. 35 (1913), 229–260.CrossRefGoogle Scholar
[37]Manning, W. A.. On the primitive groups of class fifteen. Amer. J. Math. 39 (1917), 281–310.CrossRefGoogle Scholar
[38]Manning, W. A.. On the primitive groups of class fourteen. Amer. J. Math. 51 (1929), 619–652.CrossRefGoogle Scholar
[39]McLaughlin, J. E.. Some subgroups of SLn(F2). Illinois J. Math. 13 (1969), 108–115.Google Scholar
[40]Mitchell, H. H.. Determination of the ordinary and modular ternary linear groups. Trans. Amer. Math. Soc. 12 (1911), 207–242.CrossRefGoogle Scholar
[41]Mizutani, M.. The classification of primitive permutation groups of degree less than 49. Unpublished Masters thesis (referred to in [15]).Google Scholar
[42]Pogorelov, B. A.. Primitive permutation groups of small degrees (parts I and II). Algebra and Logic 19 (1980), 230–254, 278–296.CrossRefGoogle Scholar
[43]Scott, L. L.. Representations in characteristic p. In The Santa Cruz Conference on Finite Groups. Proc. Symp. Pure Math., vol. 37 (Amer. Math. Soc., 1980), pp. 318–331.Google Scholar
[44]Scott, W. R.. Group Theory (Prentice-Hall, 1964).Google Scholar
[45]Sims, C. C.. Computational methods for permutation groups. In Computational Problems in Abstract Algebra (ed. Leech, J.), (Pergamon, 1970), pp. 169–183.Google Scholar
[46]Suprunenko, D. A.. Minimal irreducible soluble linear groups of prime degree. Trans. Moscoiv Math. Soc. 29 (1973), 215–226.Google Scholar
[47]Wagner, A.. The subgroups of PSL(5,2a). Resultate Math. 1 (1978), 207–226.CrossRefGoogle Scholar
[48]Ward, H. N.. On Ree's series of simple groups. Trans. Amer. Math. Soc. 121 (1966), 62–89.Google Scholar
[49]Wielandt, H.. Finite Permutation Groups (Academic Press, 1964).Google Scholar
[50]Wilson, R. A.. Maximal subgroups of automorphism groups of simple groups. J. London Math. Soc. (2) 32 (1985), 460–466.CrossRefGoogle Scholar