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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

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  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.]

  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.

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  3. Lipschitz:Beweis eines Satzes aus der Theorie der Substitutionen (Acta Mathematica, vol. 10, pp. 137–144, 1878).

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  4. 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).

  5. Ed. Weyr:Zur Theorie der bilinearen Formen Monatshefte für Mathematik und Physik, vol. 1, pp. 162–236, 1890; p. 209.

  6. Rost: Untersuchungen über die allgemeinste lineare Substitution, deren Potenzen eine endliche Gruppe bilden (p. 28; Teubner, Leipzig, 1892).

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  7. Maschke:Die Reduction linearer homogener Substitutionen von endlicher Periode auf ihre kanonische Form (Mathematische Annalen, vol. 50, pp. 220–224).

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    Eliakim Hastings Moore

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Moore, E.H. 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. Math. Ann. 50, 213–219 (1898). https://doi.org/10.1007/BF01448062

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