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Mathematische Annalen

, Volume 50, Issue 2–3, pp 213–219 | Cite as

An universal invariant for finite groups of linear substitutions: with application in the theory of the canonical form of a linear substitution of finite period

  • Eliakim Hastings Moore
Article

Keywords

Finite Group Canonical Form Finite Period Linear Substitution 
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ReferencesReferences

  1. *).
    Presented July 10, 1896, to the Mathematical Club of the University of Chicago in a paper entitled:Concerning Finite Groups of Linear Homogeneous Substitutions (University Record, vol. 1, p. 276, July 24, 1896). When the theorem was communicated in September, 1896, to Professor Klein, he called my attention to the fact that it had been stated (without proof) by Loewy:Sur les formes quadratiques définies à indétérminées conjuguées de M. Hermite (Comptes rendus..., vol. 123, pp. 168–171, July 20, 1896). [Addition of Oct. 16, 1897. With respect to the theorem I — of theuniversal invariant — I refer further to the report of Klein,Ueber einen Satz aus der Theorie der endlichen (discontinuirlichen) Gruppen linearer Substitutionen beliebig vieler Veränderlicher, (Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 5, p. 57, 1896), and to the closely related investigations of Valentiner (l. c., II, pp. 89, 207, 1889) and Fuchs,Ueber eine Classe linearer homogener Differentialgleichungen (Sitzungsberichte der kgl. Preussischen Akademie der Wissenschaften zu Berlin, July 9, 1896, pp. 753–769), andRemarques sur une Note de M. Alfred Loewyintitulée: 〈Sur les formes quadratiques définies à indeterminées eonjuguées de M. Hermite 〈 (Comptes Rendus, vol. 123, pp. 289–290, August 3, 1896). Fuchs in hisRemarques makes the (improper) claim that certain results of his preceding paper establish the universal invariant theorem. In fact, however, those results involve the condition (l. c. p. 768) thatat least one substitution of the group has distinct multipliers.]Google Scholar
  2. *).
    Jordan: Mémoire sur les équations différentielles linéaires à intégrale algébrique (Journal für die Mathematik, vol. 84, pp. 89–215, 1878; p. 112). The reference sometimes given forn=3 to Hermite (Journal..., vol. 47, p. 312, 1854) is in error.Google Scholar
  3. **).
    Lipschitz:Beweis eines Satzes aus der Theorie der Substitutionen (Acta Mathematica, vol. 10, pp. 137–144, 1878).Google Scholar
  4. 1.
    Kronecker:Ueber die Composition der Systeme von n2 Grössen mit sich selbst (Sitzungsberichte der kgl. Pr. Akademie der Wissenschaften zu Berlin, pp. 1081–1088, 1890; p. 1085).Google Scholar
  5. 2.
    Ed. Weyr:Zur Theorie der bilinearen Formen Monatshefte für Mathematik und Physik, vol. 1, pp. 162–236, 1890; p. 209.Google Scholar
  6. 3.
    Rost: Untersuchungen über die allgemeinste lineare Substitution, deren Potenzen eine endliche Gruppe bilden (p. 28; Teubner, Leipzig, 1892).Google Scholar
  7. 4.
    Maschke:Die Reduction linearer homogener Substitutionen von endlicher Periode auf ihre kanonische Form (Mathematische Annalen, vol. 50, pp. 220–224).Google Scholar

Copyright information

© Springer-Verlag 1898

Authors and Affiliations

  • Eliakim Hastings Moore
    • 1
  1. 1.Chicago

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