Skip to main content
SpringerLink
Log in

Generation of Linear Groups

  • Conference paper
Book cover The Geometric Vein

Abstract

Let G be a finite, primitive subgroup of GL(V)= GL(n, D), where V is an n-dimensional vector space over the division ring D. Assume that G is generated by “nice” transformations. The problem is then to try to determine (up to GL(V)-conjugacy) all possibilities for G. Of course, this problem is very vague. But it is a classical one, going back 150 years, and yet very much alive today. The purpose of this paper is to discuss both old and new results in this area, and in particular to indicate some of its history. Our emphasis will be on especially geometric situations, rather than on representation-theoretic ones.

Supported in part by NSF Grant MCS-7903130

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Amitsur, S. A. (1955), Finite subgroups of division rings. TAMS 80, 361–386.

    Article  MathSciNet  MATH  Google Scholar 

  • Aschbacher, M. (1977), A characterization of Chevalley groups over fields of odd order. Ann. of Math. (2) 106, 353–468.

    Article  MathSciNet  MATH  Google Scholar 

  • Bagnera, G. (1905), I gruppi finiti di transforminazioni lineari dello spazio che contengono omologie. Rend. Circ. Mat. Palermo 19, 1–56.

    Article  MATH  Google Scholar 

  • Blichfeldt, H. F. (1904), On the order of linear homogeneous groups (second paper). TAMS 5, 310–325.

    MathSciNet  MATH  Google Scholar 

  • Blichfeldt, H. F. (1905), The finite, discontinuous primitive groups of collineations in four variables. Math. Ann. 60, 204–231.

    Article  MathSciNet  MATH  Google Scholar 

  • Blichfeldt, H. F. (1907), The finite discontinuous primitive groups of collineations in three variables. Math. Ann. 63, 552–572.

    Article  MathSciNet  MATH  Google Scholar 

  • Blichfeldt, H. F. (1917), Finite Collineation Groups. University of Chicago Press, Chicago.

    Google Scholar 

  • Bloom, D. M. (1967), The subgroups of PSL(3,q) for odd q. TAMS 111, 150–178.

    Google Scholar 

  • Bourbaki, N. (1968), Groupes et Algèbres de Lie, Chapters IV, V, VI. Hermann, Paris.

    Google Scholar 

  • Brauer, R. (1967), Über endliche lineare Gruppen von Primzahlgrad. Math. Ann. 169, 73–96.

    Article  MathSciNet  MATH  Google Scholar 

  • Burnside, W. (1894), On a class of groups defined by congruences. Proc. LMS 25, 113–139.

    MATH  Google Scholar 

  • Burnside, W. (1895), On a class of groups defined by congruences. Proc. LMS 26, 58–106.

    MATH  Google Scholar 

  • Burnside, W. (1912), The determination of all groups of rational linear substitutions of finite order which contain the symmetric group in the variables. Proc. LMS (2) 10, 284–308.

    Article  Google Scholar 

  • Cameron, P. J. and Kantor, W. M. (1979), 2-transitive and antiflag transitive collineation groups of finite projective spaces. J. Algebra, 60, 384–422.

    Article  MathSciNet  MATH  Google Scholar 

  • Carter, R. W. (1972), Simple Groups of Lie Type. Wiley, London-New York-Sydney-Toronto.

    MATH  Google Scholar 

  • Chevalley, C. (1955), Sur certains groupes simples. Tohoku Math. J. 7, 14–66.

    Article  MathSciNet  MATH  Google Scholar 

  • Cohen, A. M. (1976), Finite complex reflection groups. Ann. Scient. Ec. Norm. Sup. (4) 9, 379–436.

    MATH  Google Scholar 

  • Cohen, A. M. (1980), Finite quaternionic reflection groups. J. Algebra 64, 293–324.

    Article  MathSciNet  MATH  Google Scholar 

  • Conway, J. H. (1971), Three lectures on exceptional groups. In Finite Simple Groups, edited by M. B. Powell and G. Higman. Academic Press, New York, pp. 215-247.

    Google Scholar 

  • Cooperstein, B. N. (1978), Minimal degree for a permutation representation of a classical group. Israel J. Math. 30, 213–235.

    Article  MathSciNet  MATH  Google Scholar 

  • Cooperstein, B. N. (1979), The geometry of root subgroups in exceptional groups I, Geom. Dedicata 8, 317–381.

    Article  MathSciNet  MATH  Google Scholar 

  • Cooperstein, B. N. (1981), Subgroups of exceptional groups of Lie type generated by long root elements, I, II. J. Algebra 70, 270–282, 283–298.

    Article  MathSciNet  MATH  Google Scholar 

  • Coxeter, H. S. M. (1948), Regular Polytopes. Methuen, London.

    MATH  Google Scholar 

  • Coxeter, H. S. M. (1957), Groups generated by unitary reflections of period two. Can. J. Math. 9, 243–272.

    Article  MathSciNet  MATH  Google Scholar 

  • Coxeter, H. S. M. (1967), Finite groups generated by unitary reflections. Abh. Hamburg 31, 125–135.

    Article  MathSciNet  MATH  Google Scholar 

  • Coxeter, H. S. M. (1974), Regular Complex Polytopes. Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  • Coxeter, H. S. M. and Todd, J. A. (1953), An extreme duodenary form. Can. J. Math. 5, 384–392.

    Article  MathSciNet  MATH  Google Scholar 

  • Curtis, C. W., Kantor, W. M., and Seitz, G. M. (1976), The 2-transitive permutation representations of the finite Chevalley groups. TAMS 218, 1–59.

    MathSciNet  MATH  Google Scholar 

  • Delsarte, P., Goethals, J. M., and Seidel, J. J. (1975), Bounds for systems of lines, and Jacobi polynomials. Philips Res. Rep. 30, 91–105.

    MATH  Google Scholar 

  • Delsarte, P., Goethals, J. M., and Seidel, J. J. (1977), Spherical codes and designs. Geom. Ded. 6, 363–388.

    Article  MathSciNet  MATH  Google Scholar 

  • Dempwolff, U. (1978), Some subgroups of GL(n,2) generated by involutions, I. J. Algebra 54, 332–352.

    Article  MathSciNet  MATH  Google Scholar 

  • Dempwolff, U. (1979), Some subgroups of GL(n,2) generated by involutions, II. J. Algebra 56, 255–261.

    Article  MathSciNet  MATH  Google Scholar 

  • Dickson, L. E. (1900), Linear Groups, with an Exposition of the Galois Field Theory. Reprinted, Dover, New York, 1958.

    MATH  Google Scholar 

  • Dickson, L. E. (1905), Determination of the ternary modular linear groups. Amer. J. Math. 27, 189–202.

    Article  MathSciNet  MATH  Google Scholar 

  • DiMartino, L. and Wagner, A. (1981), The irreducible subgroups of PSL(V 5,q), where q is odd. Resultate der Math.

    Google Scholar 

  • Doro, S. (1975), On finite linear groups in nine and ten variables. Thesis, Yale University.

    Google Scholar 

  • Feit, W. (1976), On finite linear groups in dimension at most 10. In Proc. Conf Finite Groups, edited by W. R. Scott and F. Gross. Academic Press, New York, pp. 397–407.

    Google Scholar 

  • Fischer, B. (1969), Finite groups generated by 3-transpositions. Preprint. University of Warwick.

    Google Scholar 

  • Galois, E. (1846), Oeuvres mathématiques, J. de Math. 11, 381–444.

    Google Scholar 

  • Gierster, J. (1881), Die Untergruppen der Galois’schen Gruppe der Modulargleichungen für den Fall eine primzahligen Transformationsgrades. Math. Ann. 18, 319–365.

    Article  MathSciNet  Google Scholar 

  • Hall, Jr., M. and Wales, D. B. (1968), The simple group of order 604,800. J. Algebra 9, 417–450.

    Article  MathSciNet  MATH  Google Scholar 

  • Hamill, C. M. (1951), On a finite group of order 6,531,840. Proc. LMS (2) 52, 401–454.

    Article  MathSciNet  MATH  Google Scholar 

  • Hartley, E. M. (1950), Two maximal subgroups of a collineation group in five dimensions. Proc. Camb. Phil. Soc. 46, 555–569.

    Article  MathSciNet  MATH  Google Scholar 

  • Hartley, R. W. (1926), Determination of the ternary linear collineation groups whose coefficients lie in the GF(2n). Ann. of Math. (2) 27, 140–158.

    Article  MathSciNet  Google Scholar 

  • Hazewinkel, M., Hesselink, W., Siersma, D., and Veldkamp, F. D. (1977), The ubiquity of Coxeter-Dynkin diagrams (an introduction to the A-D-E problem). Nieuw. Archief (3) 25, 257–307.

    MathSciNet  MATH  Google Scholar 

  • Ho, C. Y. (1976), Chevalley groups of odd characteristic as quadratic pairs. J. Algebra 41, 202–211.

    Article  MathSciNet  MATH  Google Scholar 

  • Hoggar, S. G. (1978), Bounds for quaternionic line systems and reflection groups. Math. Scand. 43, 241–249.

    MathSciNet  MATH  Google Scholar 

  • Huffman, W. C. (1975), Linear groups containing an element with an eigenspace of codimension two. J. Algebra 34, 260–287.

    Article  MathSciNet  MATH  Google Scholar 

  • Huffman, W. C. and Wales, D. B. (1975), Linear groups of degree n containing an element with exactly n — 2 equal eigenvalues. Linear and Multilinear Algebra 3, 53–59.

    Article  MathSciNet  Google Scholar 

  • Huffman, W. C. and Wales, D. B. (1976), Linear groups of degree eight with no elements of order 7. Ill. J. Math. 20, 519–527.

    MathSciNet  MATH  Google Scholar 

  • Huffman, W. C. and Wales, D. B. (1977), Linear groups containing an involution with two eigenvalues — 1. J. Algebra 45, 465–515.

    Article  MathSciNet  MATH  Google Scholar 

  • Huffman, W. C. and Wales, D. B. (1978), Linear groups of degree nine with no elements of order seven. J. Algebra 51, 149–163.

    Article  MathSciNet  MATH  Google Scholar 

  • Huppert, B. (1967), Endliche Gruppen I. Springer, Berlin-Heidelberg-New York.

    MATH  Google Scholar 

  • Jordan, C. (1870), Traité des Substitutions et des Équations Algébriques. Gauthier-Villars, Paris.

    Google Scholar 

  • Jordan, C. (1878), Mémoire sur les équations différentielles linéaires à intégrate algébrique. J. Reine Angew. Math. 84, 89–215.

    Article  Google Scholar 

  • Jordan, C. (1879), Sur la détermination des groupes d’ordre fini contenus dans le group linéaire. Atti Accad. Napoli 8, 1–41.

    Google Scholar 

  • Kantor, W. M. (1979a), Subgroups of classical groups generated by long root elements. TAMS 248, 347–379.

    Article  MathSciNet  MATH  Google Scholar 

  • Kantor, W. M. (1979b), Permutation representations of the finite classical groups of small degree or rank. J. Algebra 60, 158–168.

    Article  MathSciNet  MATH  Google Scholar 

  • Kantor, W. M. (1980), Linear groups containing a Singer cycle. J. Algebra 62, 232–234.

    Article  MathSciNet  MATH  Google Scholar 

  • Kantor, W. M., and Liebler, R. A. (1982), The rank 3 permutation representations of the finite classical groups. TAMS (to appear).

    Google Scholar 

  • Klein, F. (1876), Ueber binare Formen mit linearen Transformationen in sich selbst. Math. Ann. 9, 183–208.

    Article  Google Scholar 

  • Klein, F. (1879), Ueber die Transformationen siebenter Ordnung der elliptischen Functionen. Math. Ann. 14, 428–471.

    Article  Google Scholar 

  • Klein, F. (1884), Vorlesungen über das Jkosaeder und die Auflösung der Gleichungen vom fünften Grade. Teubner, Leipzig.

    Google Scholar 

  • Lindsay, II, J. H., (1971), Finite linear groups of degree six. Can. J. Math. 23, 771–790.

    Article  Google Scholar 

  • McLaughlin, J. (1967), Some groups generated by transvections. Arch. Math. 18, 364–368.

    Article  MathSciNet  MATH  Google Scholar 

  • McLaughlin, J. (1969), Some subgroups of SL n(F 2). Ill. J. Math. 13, 108–115.

    MathSciNet  MATH  Google Scholar 

  • McLaughlin, J. (1969), A simple group of order 898,128,000. In Theory of Finite Groups, edited by R. Brauer and C.-H. San. Benjamin, New York, pp. 109–111.

    Google Scholar 

  • Mitchell, H. H. (191la), Determination of the ordinary and modular ternary linear groups. TAMS 12, 207–242.

    Google Scholar 

  • Mitchell, H. H. (1911b), Note on collineation groups. Bull. AMS 18, 146–147.

    Article  MATH  Google Scholar 

  • Mitchell, H. H. (1913), Determination of the finite quaternary linear groups. TAMS 14, 123–142.

    Article  MATH  Google Scholar 

  • Mitchell, H. H. (1914a), Determination of all primitive collineation groups in more than four variables which contain homologies. Amer. J. Math. 36, 1–12.

    Article  MathSciNet  MATH  Google Scholar 

  • Mitchell, H. H. (1914b), The subgroups of the quaternary abelian linear groups. TAMS 15, 379–396.

    Article  MATH  Google Scholar 

  • Mitchell, H. H. (1935), Linear groups and finite geometries. Amer. Math. Monthly 42, 592–603.

    Article  MathSciNet  Google Scholar 

  • Moore, E. H. (1904), The subgroups of the generalized finite modular group. Decennial Publications of the University of Chicago 9, 141–190.

    Google Scholar 

  • Mwene, B. (1976), On the subgroups of the group PSL 4(2m). J. Algebra 41, 79–107.

    Article  MathSciNet  MATH  Google Scholar 

  • Mwene, B. (1982), On some subgroups of PSL(4,q) q odd.

    Google Scholar 

  • Odlyzko, A. M. and Sloane, N. J. A. (1979), New bounds on the number of unit spheres that can touch a unit sphere in n dimensions, J. Comb. Theory (A) 26, 210–214.

    Article  MathSciNet  MATH  Google Scholar 

  • Patton, W. H. (1972), The minimum index for subgroups in some classical groups: a generalization of a theorem of Galois. Thesis, University of Illinois at Chicago Circle.

    Google Scholar 

  • Piper, F. C. (1965), Collineation groups containing elations, I. Math. Z. 89, 181–191.

    Article  MathSciNet  MATH  Google Scholar 

  • Piper, F. C. (1966a), Collineation groups containing elations, II. Math. Z. 92, 281–287.

    Article  MathSciNet  MATH  Google Scholar 

  • Piper, F. C. (1966b), On elations of finite projective spaces of odd order. JLMS 41, 641–648.

    MathSciNet  MATH  Google Scholar 

  • Piper, F. C. (1968), On elations of finite projective spaces of even order. JLMS 43, 459–464.

    MathSciNet  MATH  Google Scholar 

  • Piper, F. C. (1973), On elations of finite projective spaces. Geom. Ded. 2, 13–27.

    Article  MathSciNet  MATH  Google Scholar 

  • Seitz, G. M. (1979), Subgroups of finite groups of Lie type. J. Algebra 61, 16–27.

    Article  MathSciNet  MATH  Google Scholar 

  • Seitz, G. M. (1982), Generation of finite groups of Lie type. TAMS (to appear).

    Google Scholar 

  • Serezkin, V. N. (1976), Reflection groups over finite fields of characteristic p > 5. Soviet Math. Dokl. 17, 478–480. (Correction. Dokl. Akad. Nauk SSSR 237 (1977), 504 (Russian).)

    MATH  Google Scholar 

  • Shephard, G. C. (1952), Regular complex polytopes. Proc. LMS (3) 2, 82–97.

    Article  MathSciNet  MATH  Google Scholar 

  • Shephard, G. C. (1953), Unitary groups generated by reflections. Can. J. Math. 5, 364–383.

    Article  MathSciNet  MATH  Google Scholar 

  • Shephard, G. C. and Todd, J. A. (1954), Finite unitary reflection groups. Can. J. Math. 6, 274–304.

    Article  MathSciNet  MATH  Google Scholar 

  • Singer, J. (1938), A theorem in finite projective geometry and some applications to number theory. TAMS 43, 377–385.

    Article  Google Scholar 

  • Thompson, J. G. (1970), Quadratic pairs. Actes Cong. Intern. Math. 1, 375–376.

    Google Scholar 

  • Tits, J. (1966), Classification of algebraic semi-simple groups. AMS Proc. Symp. Pure Math. 9, 33–62.

    MathSciNet  Google Scholar 

  • Tits, J. (1974), Buildings of Spherical Type and Finite BN-pairs. Springer Lecture Notes 386.

    Google Scholar 

  • Valentiner, H. (1889), De endelige Transformations-Gruppers Theori. Kjób. Skrift. (6) 5, 64–235.

    Google Scholar 

  • van der Waerden, B. L. (1935), Gruppen von Linearen Transformationen. Springer, Berlin. Reprinted: Chelsea, New York, 1948.

    MATH  Google Scholar 

  • Wagner, A. (1974), Groups generated by elations. Abh. Hamburg 41, 190–205.

    Article  MATH  Google Scholar 

  • Wagner, A. (1978), Collineation groups generated by homologies of order greater than 2. Geom. Ded. 7, 387–398.

    Article  MATH  Google Scholar 

  • Wagner, A. (1979), The subgroups of PSL(5,2a). Resultate der Math. 1, 207–226.

    Google Scholar 

  • Wagner, A. (1980-1981), Determination of finite primitive reflections groups over an arbitrary field of characteristic not two, I, II, III Geom. Dedicata 9, 239–253; 10, 191–203, 475–523.

    Article  MathSciNet  MATH  Google Scholar 

  • Wales, D. B. (1969), Finite linear groups of degree seven I. Can. J. Math. 21, 1042–1056.

    Article  MathSciNet  MATH  Google Scholar 

  • Wales, D. B. (1970), Finite linear groups of degree seven II, Pacif J. Math. 34, 207–235.

    MathSciNet  MATH  Google Scholar 

  • Wales, D. B. (1978), Linear groups of degree n containing an involution with two eigenvalues — 1, II. J. Algebra 53, 58–67.

    Article  MathSciNet  MATH  Google Scholar 

  • Wiman, A. (1896), Ueber eine einfache Gruppe von 360 ebenen Collineationen. Math. Ann. 47, 531–556.

    Article  MathSciNet  MATH  Google Scholar 

  • Wiman, A. (1899a), Bestimmung alle Untergruppen einer doppelt unendlichen Reihe von einfachen Gruppen. Bihang till K. Svenska Vet.-Akad. Handl 25, 1–47.

    Google Scholar 

  • Wiman, A. (1899b), Endliche Gruppen linearer Substitutionen. IB3f in Encyk. Math. Wiss.

    Google Scholar 

  • Zalesskii, A. E. (1976), A classification of the finite irreducible linear groups of degree 5 over a field of characteristic other than 0,2,3,5 (Russian). Dokl. Akad. Nauk BSSR 20, 773–775, 858.

    MathSciNet  MATH  Google Scholar 

  • Zalesskii, A. E. (1977), Classification of finite linear groups of degree 4 and 5 over fields of characteristic 2. Dokl. Akad. Nauk BSSR 21, 389–392, 475 (Russian).

    MathSciNet  MATH  Google Scholar 

  • Zalesskii, A. E. and Serezkin, V. N. (1976), Linear groups generated by transvections. Math. USSR Izvestija 10, 25–46.

    Article  Google Scholar 

  • Zalesskii, A. E. and Serezkin, V. N. (1977), Linear groups which are generated by pseudoreflections. Izv. Akad. Nauk BSSR, 9–16 (Russian).

    Google Scholar 

  • Zalesskii, A. E. and Serezkin, V. N. (1980), Finite linear groups generated by reflections. Izv. Akad. Nauk SSSR 44, 1279–1307 (Russian).

    MathSciNet  Google Scholar 

  • Zalesskii, A. E. and Suprenenko, I. D. (1978), Classification of finite irreducible linear groups of degree 4 over a field of characteristic p > 5. Izv. Akad. Nauk BSSR, 1978, 9–15 (Russian).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Oregon, Eugene, OR, 97403, USA

    William M. Kantor

Authors
  1. William M. Kantor
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, M5S 1A1, Canada

    Chandler Davis  & F. A. Sherk  & 

  2. Department of Mathematics, University of Washington, Seattle, WA, 98195, USA

    Branko Grünbaum

Rights and permissions

Reprints and permissions

Copyright information

© 1981 Springer-Verlag New York Inc.

About this paper

Cite this paper

Kantor, W.M. (1981). Generation of Linear Groups. In: Davis, C., Grünbaum, B., Sherk, F.A. (eds) The Geometric Vein. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-5648-9_33

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-5648-9_33

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-5650-2

  • Online ISBN: 978-1-4612-5648-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation