Abstract
The representation theory of finite groups began with the pioneering research of Frobenius, Burnside, and Schur at the turn of the century. Their work was inspired in part by two largely unrelated developments which occurred earlier in the nineteenth century. The first was the awareness of characters of finite abelian groups and their application by some of the great nineteenth-century number theorists. The second was the emergence of the structure theory of finite groups, beginning with Galois’ brief outline of the main ideas in the famous letter written on the eve of his death and continuing with the work of Sylow and others, including Frobenius himself.
This article is dedicated to the memory of my friend and collaborator, Irving Reiner.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. Brauer, Über die Darstellungen von Gruppen in Galoischen Feldern, Actualités Scientifiques et Industrielles 195, Hermann, Paris, 1935.
R. Brauer, “On the representation of a group of order g in the field of gth roots of unity,” Amer. J. Math. 67 (1945), 461–471.
R. Brauer, “On Artin’s L-series with general group characters,” Ann. of Math. (2)48 (1947), 502–514.
R. Brauer and C. J. Nesbitt, “On the modular representations of groups of finite order I,” Univ. of Toronto Studies, Math. Ser. 4, 1937.
W. Burnside, Theory of Groups of Finite Order, Cambridge, 1897; Second Edition, Cambridge, 1911.
R. Dedekind, “Zur Theorie der aus n Haupteinheiten gebildeten complexen Grössen,” Göttingen Nachr. (1885), 141–159.
L. E. Dickson, “On the group defined for any given field by the multiplication table of any given finite group,” Trans. A.M.S. 3 (1902), 285–301.
L. E. Dickson, “Modular theory of group matrices,” Trans. A.M.S. 8 (1907), 389–398.
L. E. Dickson, “Modular theory of group characters,” Bull. A.M.S. 13 (1907), 477–488.
P. G. Lejeune Dirichlet, “Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält,” Abh. Akad. d. Wiss. Berlin (1837), 45–81. Werke I, 313–342.
P. G. Lejeune Dirichlet, Vorlesungen ĂĽber Zahlentheorie, 4th ed. Published and supplemented by R. Dedekind, Vieweg, Braunschweig, 1894.
W. Feit, The Representation Theory of Finite Groups, North-Holland, Amsterdam, 1982.
W. Feit, M. Hall, and J. G. Thompson, “Finite groups in which the centralizer of any non-identity element is nilpotent,” Math. Z. 74 (1960), 1–17.
W. Feit and J. G. Thompson, “Solvability of groups of odd order,” Pacific J. Math. 13 (1963), 775–1029.
F. G. Frobenius, “Über vertauschbare Matrizen,” S’ber. Akad. Wiss. Berlin (1896), 601–614; Ges. Abh. II, 705–718.
F. G. Frobenius,“Über Gruppencharaktere ” S’ber. Akad. Wiss. Berlin (1896), 985–1021; Ges. Abh. Ill, 1–37.
F. G. Frobenius, “Über die Primfactoren der Gruppendeterminante,” S’ber. Akad. Wiss. Berlin (1896), 1343–1382; Ges. Abh. Ill, 38–77.
F. G. Frobenius, “Über die Darstellung der endlichen Gruppen durch lineare Substitutionen,” S’ber. Akad. Wiss. Berlin (1897), 994–1015; Ges. Abh. III, 82–103.
F. G. Frobenius, “Über Relationen zwischen den Charakteren einer Gruppe und denen iher Untergruppen,” S’ber. Akad. Wiss. Berlin (1898), 501–515; Ges. Abh. III, 104–118.
F. G. Frobenius, “Über den Charaktere der symmetrischen Gruppe,” S’ber. Akad. Wiss. Berlin (1900), 516–534; Ges. Abh. III, 148–166.
F. G. Frobenius, “Über die Charaktere der alternirenden Gruppe,” S’ber. Akad. Wiss. Berlin (1901), 303–315; Ges. Abh. III, 167–179.
F. G. Frobenius, “Theorie der hyperkomplexen Grössen,” S’ber. Akad. Wiss. Berlin (1903), 504–537; Ges. Abh. III, 284–317.
C. F. Gauss, Disquisitiones Arithmeticae, Leipzig, 1801; English translation by A. A. Clarke, Yale University Press, New Haven, 1966.
T. Hawkins, “The origins of the theory of group characters,” Archive Hist. Exact Sc. 7 (1971), 142–170.
T. Hawkins, “New light on Frobenius’ creation of the theory of group characters,” Archive Hist. Exact Sc. 12 (1974), 217–243.
T. Hawkins, “Hypercomplex numbers, Lie groups, and the creation of group representation theory,” Archive Hist. Exact Sc. 8 (1971), 243–287.
K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, New York, 1980.
N. Jacobson, Introduction, in EmmyNoether, Ges. Abh., Springer-Verlag, Berlin, 1983; 12–26.
W. Ledermann, “The origin of group characters,” J. Bangladesh Math. Soc. 1 (1981), 35–43.
W. Ledermann, “Issai Schur and his school in Berlin,” Bull. London Math. Soc. 15 (1983), 97–106.
A. Loewy, “Sur les formes quadratiques définies à indéterminées conjuguées de M. Hermite,” Comptes Rendus Acad. Sei. Paris 123 (1896), 168–171.
H. Maschke, “Beweis des Satzes, dass diejenigen endlichen linearen Substitutionsgruppen, in welchen einige durchgehends verschwindende Coefficienten auftreten, intransitiv sind,” Math. Ann. 52 (1899), 363–368.
E. H. Moore, “A universal invariant for finite groups of linear substitutions: with applications in the theory of the canonical form of a linear substitution of finite period,” Math. Ann. 50 (1898), 213–219.
E. Noether, “Hyperkomplexe Grössen und Darstellungstheorie,” Math. Z. 30 (1929), 641–692; Ges. Abh. 563–992.
I. Schur, Über eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen, Dissertation, Berlin, 1901; Ges. Abh. 1, 1–72.
I. Schur, “Neue Begründung der Theorie der Gruppencharaktere,” S’ber. Akad. Wiss. Berlin (1905), 406–432; Ges. Abh. I, 143–169.
I. Schur, “Arithmetische Untersuchungen über endliche Gruppen linearer Substitutionen” S’ber. Akad. Wiss. Berlin (1906), 164–184; Ges. Abh. I, 177–197.
I. Schur, “Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen,” J, reine u. angew. Math. 132 (1907), 85–137; Ges. Abh. I, 198–205.
E. Study, “Über Systeme von complexen Zahlen,” Göttingen Nach. (1889), 237–268.
M. Suzuki, “The non-existence of a certain type of simple group of odd order,” Proc. A.M.S. 8 (1957), 686–695.
H. Weber, Lehrbuch der Algebra, vol. 2, Vieweg, Braunschweig, 1896.
K. Weierstrass, “Zur Theorie der aus n Haupteinheiten gebildeten complexen Grössen,” Göttingen Nach. (1884), 395–414.
A. Weil, “Numbers of solutions of equations in finite fields,” Bull. A.M.S. 55 (1949), 497–508; Collected Papers, I, 399–410.
Rights and permissions
Copyright information
© 2001 Springer Science+Business Media New York
About this chapter
Cite this chapter
Curtis, C.W. (2001). Representation Theory of Finite Groups: from Frobenius to Brauer. In: Mathematical Conversations. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0195-0_12
Download citation
DOI: https://doi.org/10.1007/978-1-4613-0195-0_12
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-6556-6
Online ISBN: 978-1-4613-0195-0
eBook Packages: Springer Book Archive