The Mathematical Intelligencer

, Volume 13, Issue 1, pp 7–11 | Cite as

Years ago

A study in group theory: Leonard eugene dickson’sLinear Groups


Simple Group Finite Field Linear Group Karen Finite Simple Group 
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  1. 1.
    Leonard Eugene Dickson,Linear Groups with an Exposition of the Galois Field Theory (Leipzig: B. G. Teubner, 1901; reprint ed., New York: Dover Publications, Inc., 1958). Dickson’s thesis appeared as “The Analytic Representation of Substitutions on a Power of a Prime Number of Letters with a Discussion of the Linear Group,”Annals of Mathematics 11 (1897), 65–120, 161–183, orThe Collected Mathematical Papers of Leonard Eugene Dickson, (A. Adrian Albert ed.) 5 vols., New York: Chelsea Publishing Co. (1975), 2: 651–729.zbMATHGoogle Scholar
  2. 2.
    Wilhelm Magnus, Introduction to the Dover edition,Linear Groups, p. v.Google Scholar
  3. 3.
    Several biographical sources on Dickson exist, but see, for example, A. Adrian Albert, “Leonard Eugene Dickson 1874-1954,”Bulletin of the American Mathematical Society 61 (1955), 331–345; Raymond Clare Archibald, ed.,A Semicentennial History of the American Mathematical Society 1888–1938, 2 vols., New York: American Mathe- matical Society (1938); reprint ed., New York: Arno Press (1980), 1, 183–194; Charles C. Gillispie, ed.,The Dictionary of Scientific Biography, 16 vols., 1 supp., New York: Charles S. Scribner’s Sons (1970–1980),s.v. “Dickson, Leonard Eugene,” by Ronald S. Calinger.CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    D. Reginald Traylor with William Bane and Madeline Jones,Creative Teaching: Heritage of R. L. Moore, Houston: University of Houston (1972), 28.Google Scholar
  5. 5.
    On the atmosphere at Chicago and in its Mathematics Department at this time, see Karen Hunger Parshall, “Eliakim Hastings Moore and the founding of a mathematical community in America, 1892–1902,“Annals of Science 41 (1984), 313–333; reprinted in Peter Duren,et al., ed.,A Century of Mathematics in America—Part II, Providence: American Mathematical Society (1988), 155–175. The developments at Chicago are viewed from the broader perspective of late nineteenth-century mathematical developments in America in Karen Hunger Parshall and David E. Rowe,The Emergence of an American Mathematical Research Community: J. J. Sylvester, Felix Klein, and E. H. Moore, forthcoming in the joint American Mathematical Society/London Mathematical Society series in the History of Mathematics.CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Eliakim Hastings Moore, “A doubly-infinite system of simple groups,”Mathematical Papers Read at the International Mathematical Congress Held in Connection with the World’s Columbian Exposition: Chicago 1893 (E. H. Moore, Oskar Bolza, Heinrich Maschke, and Henry S. White, ed.), New York: Macmillan & Co. (1896), 208–242. (Hereinafter cited asCongress Papers.)Google Scholar
  7. 7.
    Felix Klein, “Über die Transformation der elliptischen Funktionen und die Auflösung der Gleichungen fünften Grades,“Mathematische Annalen 14 (1879), 111–172.CrossRefzbMATHGoogle Scholar
  8. 8.
    E. H. Moore, “Concerning a congruence group of order 360 contained in the group of linear fractional substitutions,”Proceedings of the American Association for the Advancement of Science 41 (1892), 62; and Frank N. Cole, “On a certain simple group,“ 40–43 inCongress Papers.Google Scholar
  9. 9.
    Moore, “A doubly-infinite system of simple groups,” 238–242.Google Scholar
  10. 10.
    Ibid., 211.Google Scholar
  11. 11.
    Dickson, “The analytic representation of substitutions,” 68–120 or 652–706. Our notation for the prime changes here in order to conform to Dickson’s usage.Google Scholar
  12. 12.
    Camille Jordan,Traité des Substitutions et des Équations algébriques, Paris: Gauthier-Villars (1870).Google Scholar
  13. 13.
    Dickson, “The analytic representation of substitutions,” 135 or 721.Google Scholar
  14. 14.
    Ibid., 67 or 653.Google Scholar
  15. 15.
    In a sample space of 320 “active“ members of the American mathematical community in the years from 1891 to 1906, 112 or 35.0% reported spending some time studying abroad. See Delia Dumbaugh and Karen Hunger Parshall, “A profile of the American mathematical research community: 1891–1906,“ to appear.Google Scholar
  16. 16.
    Leonard E. Dickson to E. H. Moore, December 19, 1899, University of Chicago Archives, E. H. Moore Papers, Box 1, Folder 19. As always, I thank the University of Chicago for permission to quote from its archives.Google Scholar
  17. 17.
    Leonard E. Dickson to Felix Klein, April 7, 1900, Klein Nachlaß VIII, Archive 528/1, Niedersächische Staatsund Universitätsbibliothek (NSUB), Göttingen. I thank the library for permission to quote from its archives and Dr. Helmut Rohlfing, the director of the library’s Handschriftenabteilung, for his help and hospitality during my recent research trip to Göttingen. 18. Leonard E. Dickson to Felix Klein, April 7, 1900, Klein Nachlaß VIII, Archive 528/2, NSUB, Göttingen. Dickson gave his simplified treatment of this point inLinear Groups, 208–216.Google Scholar
  18. 19.
    Dickson,Linear Groups, ix.Google Scholar
  19. 20.
    William Burnside,Theory of Groups of Finite Order, Cambridge: University Press (1911); reprint ed., New York: Dover Publications, Inc. (1955), viii. The preface to the first edition was reprinted in the second.zbMATHGoogle Scholar
  20. 21.
    Among the pertinent works by Frobenius, see Georg Frobenius, “Ueber Gruppencharaktere,“Sitzungsberichte der Preußischen Akademie der Wissenschaften zu Berlin (1896), 985–1021; “Ueber die Primfactoren der Gruppendeterminante,”op. cit., 1343–1382; “Ueber die Darstellung der endlichen Gruppen durch lineare Substitutionen,“op. cit. (1897), 994–1015; “Ueber die Darstellung der endlichen Gruppen durch lineare Substitutionen II,“op. cit. (1899), 482–500; and “Ueber die Composition der Charaktere einer Gruppe,“op. cit., 330–339. These works may also be found in Georg Frobenius,Gesammelte Abhandlungen, (Jean-Pierre Serre ed.) 3 vols., Berlin: Springer-Verlag (1968).Google Scholar
  21. 22.
    Burnside, v.Google Scholar
  22. 23.
    For a masterful historical treatment of Frobenius and his work, see Thomas Hawkins, “The origins of the theory of group characters,“Archive for History of Exact Sciences 7 (1971), 142–171; and “New light on Frobenius’ creation of the theory of group characters,”op. cit. 12 (1974), 217–243.CrossRefMathSciNetGoogle Scholar
  23. 24.
    For each of the classical linear groups (that is, the general and special linear, the unitary, the symplectic, and the orthogonal groups), Dickson presented a formula for its order. Remarkably “modern,” his order formulas were displayed in a form suggestive of those for the finite Chevalley groups, based on the Bruhat decomposition and involving the exponents of the Weyl groups. See, for instance, Roger W. Carter,Simple Groups of Lie Type, New York: John Wiley & Sons (1972). I thank my resident expert in algebraic groups, Brian J Parshall, for pointing this out to me.zbMATHGoogle Scholar
  24. 25.
    Fifty years later, Jean Dieudonné showed that Dickson’s list of isomorphisms was, in fact, complete. See Jean Dieudonné,On the Automorphisms of the Classical Groups with a Supplement by Loo-Keng Hua, vol. 2,Memoirs of the American Mathematical Society (1951).Google Scholar
  25. 26.
    Dickson,Linear Groups, 308–310.Google Scholar
  26. 27.
    Ibid., 309. As Dickson pointed out, this non-isomorphism had first been proven using brute force by the American female mathematician, Ida May Schottenfels in “Two non-isomorphic simple groups of the same order 20160,“Annals of Mathematics (2)1 (1900), 147–152. Schottenfels was one of the two women who emerged among the sixty-two “most active“ participants in the American mathematical research community between 1891 and 1906. In all, seventy-one women surfaced in a total sample space of 1061. See note 15 above, and Delia Dumbaugh and Karen Hunger Parshall, “Women in the American Mathematical Research Community: 1891–1906,“ to appear.Google Scholar
  27. 28.
    Here, I have used not Dickson’s now antiquated notation for these groups but rather that used in J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson,Atlas of Finite Simple Groups, Oxford: Clarendon Press (1985). Since these groups are all simple, they are also denoted simply byL m (q).zbMATHGoogle Scholar
  28. 29.
    Emil Artin, “The orders of the linear groups,”Communications in Pure and Applied Mathematics 8 (1955), 355–365.CrossRefzbMATHMathSciNetGoogle Scholar
  29. 30.
    Emil Artin, “The orders of the classical simple groups,” —, 455–472.CrossRefzbMATHMathSciNetGoogle Scholar
  30. 31.
    Dickson,Linear Groups, 303–307.Google Scholar

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© Springer Science+Business Media, Inc. 1991

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