Advertisement

Classical Groups

  • Erich W. Ellers
Chapter
Part of the NATO ASI Series book series (ASIC, volume 333)

Abstract

If a group has a set of generators with outstanding properties, then the factorization of group elements into generators will provide information on the structure of the group. It is advantageous to determine the minimal number of factors needed to express an element as a product of generators. This number is called the length of a group element. The Cartan-Dieudonné theorem is a well-known example for results of this kind.

The classical groups have distinguished sets of generators. The general linear group is generated by simple mappings, the orthogonal group by reflections, the symplectic group by transvections, the unitary group by quasireflections, the group of projectivities by dilatations, the group of equiaffinities by translations and shears. The orthogonal group yields a second outstanding set of generators, namely the set of all orthogonal involutions.

We shall report on the solution of the length problem for a number of classical groups. We shall discuss whenever possible different generating sets and the resulting difference in the length of an element.

Keywords

Conjugacy Class Simple Mapping Orthogonal Group Symplectic Group Symmetric Bilinear Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Artin, E. ”Geometric Algebra”, Interscience Publishers, New York, 1957.zbMATHGoogle Scholar
  2. 2.
    Ballantine, C. S. ”Some involutory similarities”, Linear and Multilinear Algebra 3 (1975), 19–23.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ballantine, C. S. ”Products of involutory matrices”, Linear and Multilinear Algebra 5 (1977), 53–62.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Baltag I. A. and Garit, V. P. ”Classification of equiaffine transformations of the Euclidean plane”, Studies in Algebra, Mathematical Analysis, and their Applications (Russian), 118. Izdat. ”Stiinca”, Kishinev (1977), 3–7.Google Scholar
  5. 5.
    Bottema, O. ”Equiaffinities in three-dimensional space”, Univ. Beograd. Publ. Elektrotehn. Fak., Ser. Mat. Fiz. 602–633 (1978), 9–15.MathSciNetGoogle Scholar
  6. 6.
    Cater, F. S. ”Products of central collineations”, Linear Algebra Appl. 19 (1978), 251–274.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Coxeter, H. S. M. ”Products of shears in an affine Pappian plane”, Rend. Mat. (6) 3, No. 1 (1970), 1–6.Google Scholar
  8. 8.
    Crilly, T. ”Finite vector spaces from rotating triangles”, Math. Mag. 52, No. 3 (1979), 163–168.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    D’Angelo, M. ”On equiaffine planes”, J. Geom. 15/1 (1981), 74–88.MathSciNetGoogle Scholar
  10. 10.
    Dennis, R. K. and Vaserstein, L. N. ”On a question of M. Newman on the number of commutators”, J. Algebra 118, No. 1 (1988), 150–161.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Dennis, R. K. and Vaserstein, L. N. ”Commutators in linear groups”, K-Theory 2 (1989), 761–767.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Dieudonné, J. ”La géométrie des groupes classiques”, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1955.zbMATHGoogle Scholar
  13. 13.
    Dieudonné, J. ”Sur les générateurs des groupes classiques”, Summa Bras. Math. 3 (1955), 149–179.Google Scholar
  14. 14.
    Djoković, D. Ž. ”Product of two involutions”, Arch. Math. (Basel) 18 (1967), 582–584.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Djoković, D. Ž. ”Products of reflections in the quaternionic unitary group”, J. Algebra 59 (1979), 399–411.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Djoković, D. Ž. ”Products of positive reflections in real orthogonal groups”, Pacific J. Math. 107, No. 2 (1983), 341–348.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Djoković, D. Ž. ”Characterization of dilatations which are expressible as a product of three transvections or three reflections”, Proc. Amer. Math. Soc. 92, No. 3 (1984), 315–319.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Djoković, D. Ž. and Malzan, J. ”Products of reflections in the general linear group over a division ring”, Linear Algebra Appl. 28 (1979), 53–62.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Djoković, D. Ž. and Malzan, J. ”Products of reflections in the unitary group”, Proc. Amer. Math. Soc. 73, No. 2 (1979), 157–160.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Djoković, D. Ž. and Malzan, J. ”Products of reflections in the quaternionic unitary group”, J. Algebra 59 (1979), 399–411.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Djoković, D. Ž. and Malzan, J. ”Products of reflections in U(p, q)”, Mem. Amer. Math. Soc. 37, No. 259 (1982), 1–82.Google Scholar
  22. 22.
    Ellers, E. W. ”The length problem for the equiaffme group of a Pappian geometry”, Rend. Mat (6) 9, No. 2 (1976), 327–336.MathSciNetzbMATHGoogle Scholar
  23. 23.
    Ellers, E. W. ”Decomposition of orthogonal, symplectic, and unitary isome-tries into simple isometries”, Abh. Math. Sem. Univ. Hamburg 46 (1977), 97–127.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Ellers, E. W. ”Decomposition of equiaffinities into reflections”, Geom. Dedi-cata 6 (1977), 297–304.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Ellers, E. W. ”Bireflectionality in classical groups”, Canad. J. Math. 29 (1977), 1157–1162.MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Ellers, E. W. ”Relations in classical groups”, J. Algebra 51, No. 1 (1978), 19–24.MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Ellers, E. W. ”Factorization of affinities”, Canad. J. Math. 31, No. 2 (1979), 354–362.MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Ellers, E. W. ”Products of two involutory matrices over skewfields”, Linear Algebra Appl. 26 (1979), 59–63.MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Ellers, E. W. ”Relations in hyperreflection groups”, Proc. Amer. Math. Soc. 81, No. 2 (1981), 167–171.MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Ellers, E. W. ”Defining relations for the equiaffme group”, Geom. Dedicata 10 (1981), 177–182.MathSciNetCrossRefGoogle Scholar
  31. 31.
    Ellers, E. W. ”Projective collineations as products of homologies, elations, and projective reflections”, Aequationes Math. 25 (1982), 103–114. Short communication: Aequationes Math. 25, 116–117.MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Ellers, E. W. ”Products of axial affinities and products of central collineations”, The Geometric Vein (Coxeter Festschrift), Springer-Vorlag, New York (1982), 465–470.Google Scholar
  33. 33.
    Ellers, E. W. ”Relations in the projective general linear group and in the affine subgroup”, J. Algebra 77, No. 2 (1982), 333–337.MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Ellers, E. W. ”Skew affinities”, Geom. Dedicata 12 (1982), 17–24.MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Ellers, E. W. ”The Minkowski group”, Geom. Dedicata 15 (1984), 363–375.MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Ellers, E. W. ”Products of elations and harmonic homologies”, Canad. Math. Bull. 28(4) (1985), 397–400.MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Ellers, E. W. ”Products of half-turns”, J. Algebra 99, No. 2 (1986), 275–294.MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Ellers, E. W.; Frank R. and Nolte, W. ”Bireflectionality of the weak orthogonal and the weak symplectic groups”, J. Algebra 88, No. 1 (1984), 63–67.MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Ellers, E. W. and Malzan, J. ”Products of reflections in GL(n, IH)”, Linear and Multilinear Algebra 20 (1987), 281–324.MathSciNetzbMATHCrossRefGoogle Scholar
  40. 11.
    Dennis, R. K. and Vaserstein, L. N. ”Commutators in linear groups”, K-Theory 2 (1989), 761–767.MathSciNetzbMATHCrossRefGoogle Scholar
  41. 12.
    Dieudonné, J. ”La géométrie des groupes classiques”, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1955.zbMATHGoogle Scholar
  42. 13.
    Dieudonné, J. ”Sur les générateurs des groupes classiques”, Summa Bras. Math. 3 (1955), 149–179.Google Scholar
  43. 14.
    Djoković, D. Ž. ”Product of two involutions”, Arch. Math. (Basel) 18 (1967), 582–584.MathSciNetzbMATHCrossRefGoogle Scholar
  44. 15.
    Djoković, D. Ž. ”Products of reflections in the quaternionic unitary group”, J. Algebra 59 (1979), 399–411.MathSciNetzbMATHCrossRefGoogle Scholar
  45. 16.
    Djoković, D. Ž. ”Products of positive reflections in real orthogonal groups”, Pacific J. Math. 107, No. 2 (1983), 341–348.MathSciNetzbMATHCrossRefGoogle Scholar
  46. 17.
    Djoković, D. Ž. ”Characterization of dilatations which are expressible as a product of three transvections or three reflections”, Proc. Amer. Math. Soc. 92, No. 3 (1984), 315–319.MathSciNetzbMATHCrossRefGoogle Scholar
  47. 18.
    Djoković, D. Ž. and Malzan, J. ”Products of reflections in the general linear group over a division ring”, Linear Algebra Appl. 28 (1979), 53–62.MathSciNetzbMATHCrossRefGoogle Scholar
  48. 19.
    Djoković, D. Ž. and Malzan, J. ”Products of reflections in the unitary group”, Proc. Amer. Math. Soc. 73, No. 2 (1979), 157–160.MathSciNetzbMATHCrossRefGoogle Scholar
  49. 20.
    Djoković, D. Ž. and Malzan, J. ”Products of reflections in the quaternionic unitary group”, J. Algebra 59 (1979), 399–411.MathSciNetzbMATHCrossRefGoogle Scholar
  50. 21.
    Djoković, D. Ž. and Malzan, J. ”Products of reflections in U(p, q)”, Mem. Amer. Math. Soc. 37, No. 259 (1982), 1–82.Google Scholar
  51. 22.
    Ellers, E. W. ”The length problem for the equiaffme group of a Pappian geometry”, Rend. Mat (6) 9, No. 2 (1976), 327–336.MathSciNetzbMATHGoogle Scholar
  52. 23.
    Ellers, E. W. ”Decomposition of orthogonal, symplectic, and unitary isome-tries into simple isometries”, Abh. Math. Sem. Univ. Hamburg 46 (1977), 97–127.MathSciNetzbMATHCrossRefGoogle Scholar
  53. 24.
    Ellers, E. W. ”Decomposition of equiaffinities into reflections”, Geom. Dedi-cata 6 (1977), 297–304.MathSciNetzbMATHCrossRefGoogle Scholar
  54. 25.
    Ellers, E. W. ”Bireflectionality in classical groups”, Canad. J. Math. 29 (1977), 1157–1162.MathSciNetzbMATHCrossRefGoogle Scholar
  55. 26.
    Ellers, E. W. ”Relations in classical groups”, J. Algebra 51, No. 1 (1978), 19–24.MathSciNetzbMATHCrossRefGoogle Scholar
  56. 27.
    Ellers, E. W. ”Factorization of affinities”, Canad. J. Math. 31, No. 2 (1979), 354–362.MathSciNetzbMATHCrossRefGoogle Scholar
  57. 28.
    Ellers, E. W. ”Products of two involutory matrices over skewfields”, Linear Algebra Appl. 26 (1979), 59–63.MathSciNetzbMATHCrossRefGoogle Scholar
  58. 29.
    Ellers, E. W. ”Relations in hyperreflection groups”, Proc. Amer. Math. Soc. 81, No. 2 (1981), 167–171.MathSciNetzbMATHCrossRefGoogle Scholar
  59. 30.
    Ellers, E. W. ”Defining relations for the equiaffme group”, Geom. Dedicata 10 (1981), 177–182.MathSciNetCrossRefGoogle Scholar
  60. 31.
    Ellers, E. W. ”Projective collineations as products of homologies, elations, and projective reflections”, Aequationes Math. 25 (1982), 103–114. Short communication: Aequationes Math. 25, 116–117.MathSciNetzbMATHCrossRefGoogle Scholar
  61. 32.
    Ellers, E. W. ”Products of axial affinities and products of central collineations”, The Geometric Vein (Coxeter Festschrift), Springer-Vorlag, New York (1982), 465–470.Google Scholar
  62. 33.
    Ellers, E. W. ”Relations in the projective general linear group and in the affine subgroup”, J. Algebra 77, No. 2 (1982), 333–337.MathSciNetzbMATHCrossRefGoogle Scholar
  63. 34.
    Ellers, E. W. ”Skew affinities”, Geom. Dedicata 12 (1982), 17–24.MathSciNetzbMATHCrossRefGoogle Scholar
  64. 35.
    Ellers, E. W. ”The Minkowski group”, Geom. Dedicata 15 (1984), 363–375.MathSciNetzbMATHCrossRefGoogle Scholar
  65. 36.
    Ellers, E. W. ”Products of elations and harmonic homologies”, Canad. Math. Bull. 28 (4) (1985), 397–400.MathSciNetzbMATHCrossRefGoogle Scholar
  66. 37.
    Ellers, E. W. ”Products of half-turns”, J. Algebra 99, No. 2 (1986), 275–294.MathSciNetzbMATHCrossRefGoogle Scholar
  67. 38.
    Ellers, E. W.; Frank R. and Nolte, W. ”Bireflectionality of the weak orthogonal and the weak symplectic groups”, J. Algebra 88, No. 1 (1984), 63–67.MathSciNetzbMATHCrossRefGoogle Scholar
  68. 39.
    Ellers, E. W. and Malzan, J. ”Products of reflections in GL(n, IH)”, Linear and Multilinear Algebra 20 (1987), 281–324.MathSciNetzbMATHCrossRefGoogle Scholar
  69. 40.
    Ellers, E. W. and Malzan, J. ”Transvections as generators of the special linear group over the quaternions”, J. Pure Appl. Algebra 57 (1989), 25–31.MathSciNetzbMATHCrossRefGoogle Scholar
  70. 41.
    Ellers, E. W. and Malzan, J. ”Solving the length problem for the special linear group over the quaternions”, Algebras, Groups and Geometries 6 (1989), 135–152.MathSciNetzbMATHGoogle Scholar
  71. 42.
    Ellers, E. W. and Malzan, J. ”Products of reflections in the kernel of the spinorial norm”, Geom. Dedicata (1990).Google Scholar
  72. 43.
    Ellers, E. W. and Malzan, J. ”Products of positive transvections in the real symplectic group”, Comm. Algebra (1991).Google Scholar
  73. 44.
    Ellers, E. W. and Malzan, J. ”Products of λ-transvections in Sp(2n,q)”, Linear Algebra Appl. (1991).Google Scholar
  74. 45.
    Ellers, E. W. and Malzan, J. ”The length problem for the special unitary group generated by positive transvections”, (preprint).Google Scholar
  75. 46.
    Ellers, E. W. and Nolte, W. ”Radical relations in orthogonal groups”, Linear Algebra Appl. 38 (1981), 135–139.MathSciNetzbMATHCrossRefGoogle Scholar
  76. 47.
    Ellers, E. W. and Nolte, W. ”Bireflectionality of orthogonal and symplectic groups”, Arch. Math. (Basel) 39 (1982), 113–118.MathSciNetzbMATHCrossRefGoogle Scholar
  77. 48.
    Gow, R. ”Products of two involutions in classical groups of characteristic 2”, J. Algebra 71, No. 2 (1981), 583–591.MathSciNetzbMATHCrossRefGoogle Scholar
  78. 49.
    Gustafson, W. H.; Halmos, P. R. and Radjavi, H. ”Products of involutions”, Linear Algebra Appl. 13 (1976), 157–162.MathSciNetzbMATHCrossRefGoogle Scholar
  79. 50.
    Hahn, A. J. and O’Meara, O.T. ”The Classical Groups and K-Theory”, Grundlehren Math. Wiss. 291, Springer-Verlag, Berlin-Heidelberg, 1989.Google Scholar
  80. 51.
    Halmos, P. R. and Kakutani, S. ”Products of symmetries”, Bull. Amer. Math. Soc. 64 (1958), 77–78.MathSciNetzbMATHCrossRefGoogle Scholar
  81. 52.
    Hoffman, F. and Paige, E. C. ”Products of two involutions in the general linear group”, Indiana Univ. Math. J. 20, No. 11 (1971), 1017–1020.MathSciNetzbMATHCrossRefGoogle Scholar
  82. 53.
    Huppert, B. ”Isometrien von Vektorräumen I”, Arch. Math. 35 (1980), 164–176.MathSciNetzbMATHCrossRefGoogle Scholar
  83. 54.
    Ishibashi, H. ”Generators of an orthogonal group over a finite field”, Czechoslovak Math. J. 28 (103) (1978), 419–433.MathSciNetGoogle Scholar
  84. 55.
    Jacobson, N. ”The Theory of Rings”, Math. Surveys 11, Amer. Math. Soc, Providence, R. I, 1943.Google Scholar
  85. 56.
    Johnsen, E. C. ”Essentially doubly stochastic matrices — Products of elementary matrices”, Linear and Multilinear Algebra 1 (1973), 33–45.MathSciNetCrossRefGoogle Scholar
  86. 57.
    Johnsen, E. C. ”Essentially stochastic matrices — Factorizations into elementary and quasielementary matrices”, Linear Algebra Appl. 17 (1977). 79–93.MathSciNetzbMATHCrossRefGoogle Scholar
  87. 58.
    Johnsen, E. C. ”Real essentially stochastic matrices — Factorizations into special elementary matrices”, Linear and Multilinear Algebra 10 (1981), 319–328.MathSciNetzbMATHCrossRefGoogle Scholar
  88. 59.
    Knüppel, F. ”Products of involutions in orthogonal groups”, Ann. Discrete Math. 37 (1988), 231–248.CrossRefGoogle Scholar
  89. 60.
    Knüppel, F. and Nielsen, K. ”On products of two involutions in the orthogonal group of a vector space”, Linear Algebra Appl. 94 (1987), 209–216.MathSciNetzbMATHCrossRefGoogle Scholar
  90. 61.
    Knüppel, F. and Nielsen, K. ”Products of involutions in O +(V)”, Linear Algebra Appl. 94 (1987), 217–222.MathSciNetzbMATHCrossRefGoogle Scholar
  91. 62.
    Knüppel, F.; Nielsen, K. and Thomsen, G. ”Minkowski half-turns”, Geom. Dedicata 33 (1990), 77–81.MathSciNetzbMATHCrossRefGoogle Scholar
  92. 63.
    Korobov, A. A. ”Decomposition of simple transformations into simple transformations with given parameters”, Some Problems in Differential Equations and Discrete Mathematics (Russian), Novosibirsk State University, Novosibirsk (1986), 28–33.Google Scholar
  93. 64.
    Liu, K. M. ”Decomposition of matrices into three involutions”, Linear Algebra Appl. 111 (1988), 1–24.MathSciNetzbMATHCrossRefGoogle Scholar
  94. 65.
    Malzan, J. ”Products of positive reflections in the orthogonal group”, Canad. J. Math. 34 (1982), 484–499.MathSciNetzbMATHCrossRefGoogle Scholar
  95. 66.
    Marcus, M.; Kidman, K. and Sandy, M. ”Products of elementary doubly stochastic matrices”, Linear and Multilinear Algebra 15 (1984), 331–340.MathSciNetzbMATHCrossRefGoogle Scholar
  96. 67.
    Nielsen, K. ”Produkte von Involutionen und Längenprobleme in klassischen Gruppen”, Ph.D. thesis, University of Kiel, West Germany, 1987.Google Scholar
  97. 68.
    Nielsen, K. ”On bireflectionality and trireflectionality of orthogonal groups”, Linear Algebra Appl. 94 (1987), 197–208.MathSciNetzbMATHCrossRefGoogle Scholar
  98. 69.
    O’Meara, O. T. ”Lectures on Linear Groups”, CBMS Regional Conf. Ser. in Math. 22, Amer. Math. Soc, Providence, R. I., 1974.Google Scholar
  99. 70.
    O’Meara, O. T. ”Symplectic Groups”, Math. Surveys 16, Amer. Math. Soc., Providence, R. I., 1978.Google Scholar
  100. 71.
    Phadke, B. B. ”Products of transvections”, Canad. J. Math 26 (1974), 1412–1417.MathSciNetzbMATHCrossRefGoogle Scholar
  101. 72.
    Phadke, B. B. ”Products of reflections”, Arch. Math. (Basel) 26 (1975), 663–665.MathSciNetzbMATHCrossRefGoogle Scholar
  102. 73.
    Radjavi, H. ”Decomposition of matrices into simple involutions”, Linear Algebra Appl. 12 (1975), 247–255.MathSciNetzbMATHCrossRefGoogle Scholar
  103. 74.
    Schaal, H. ”Zur Perspektiven Zerlegung und Fixpunktkonstruktion der Affinitäten von A n(K)” Arch. Math. (Basel) 38 (1982), 116–123.MathSciNetzbMATHCrossRefGoogle Scholar
  104. 75.
    Schaal, H. ”Bewegungen als Scherungsprodukte”, Der Mathematikunterricht 29, Heft 6 (1983), 43–55.Google Scholar
  105. 76.
    Schaal, H. ”Zur Zerlegung der ebenen unimodularen Affinitäten in Scherungen oder Spiegelungen”, Sitzungsber. Heidelb. Akad. Wiss. Math.-Natur. Kl., Abt. II, 193, Heft 8–10 (1984), 485–500.MathSciNetzbMATHGoogle Scholar
  106. 77.
    Snapper, E. and Troyer, R. J. ”Metric Affine Geometry”, Academic Press, New York, 1971.zbMATHGoogle Scholar
  107. 78.
    Vaserstein, L. N. ”Normal subgroups of orthogonal groups over commutative rings”, Amer. J. Math. 110 (1988), 955–973.MathSciNetzbMATHCrossRefGoogle Scholar
  108. 79.
    Vaserstein, L. N. ”Normal subgroups of symplectic groups over rings”, K-Theory 2 (1989), 647–673.MathSciNetzbMATHCrossRefGoogle Scholar
  109. 80.
    Vaserstein, L. N. ”Normal subgroups of classical groups over rings and gauge groups”, Contemp. Math. 83 (1989), 451–459.MathSciNetCrossRefGoogle Scholar
  110. 81.
    Vaserstein, L. N. and Magurn, B. A. ”Prestabilization for K 1 of Banach Algebras”, Linear Algebra Appl. 95 (1987), 69–96.MathSciNetzbMATHCrossRefGoogle Scholar
  111. 82.
    Waterhouse, W. C. ”Factoring unimodular matrices, Solutions of Advanced Problems”, Amer. Math. Monthly 81, No. 9 (1974), 1035.MathSciNetGoogle Scholar
  112. 83.
    Witczyński, K. ”Projective collineations as products of cyclic collineations”, Demonstratio Math. 12, No. 4 (1979), 1111–1125.MathSciNetzbMATHGoogle Scholar
  113. 84.
    Witczyński, K. ”On generators of the group of projective transformations”, Demonstratio Math. 14, No. 4 (1981), 1053–1075.MathSciNetzbMATHGoogle Scholar
  114. 85.
    Wonenburger, M. J. ”Transformations which are products of two involutions”, J. Math. Mech. 16 (1966), 327–338.MathSciNetzbMATHGoogle Scholar
  115. 86.
    Wong, W. J. ”A theorem on generation of finite orthogonal groups”, J. Austral. Math. Soc. 16 (1973), 495–506.MathSciNetzbMATHCrossRefGoogle Scholar
  116. 87.
    Wong, W. J. ”Generators and relations for classical groups”, J. Algebra 32, No. 3 (1974), 529–553.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1991

Authors and Affiliations

  • Erich W. Ellers
    • 1
  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

Personalised recommendations