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On a theorem of Jordan
Author:
Jean-Pierre Serre
Journal:
Bull. Amer. Math. Soc. 40 (2003), 429-440
MSC (2000):
Primary 06-XX, 11-XX, 11F11
Posted:
July 17, 2003
MathSciNet review:
1997347
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Abstract: The theorem of Jordan which I want to discuss here dates from 1872. It is an elementary result on finite groups of permutations. I shall first present its translations in Number Theory and Topology.
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- 1.
- E. Artin, Über eine neue Art von L-Reihen, Hamb. Abh. 3 (1923), 89-108 (= Coll. Papers, 105-124).
- 2.
- N. Boston, W. Dabrowski, T. Foguel, P. J. Gies, D. A. Jackson, J. Leavitt and D. T. Ose, The proportion of fixed-point-free elements of a transitive permutation group, Comm. Algebra 21 (1993), 3259-3275. MR 94e:20002
- 3.
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- 4.
- P. J. Cameron and A. M. Cohen, On the number of fixed point free elements in a permutation group, Discrete Math. 106/107 (1992), 135-138. MR 93f:20004
- 5.
- T. Crespo, Galois representations, embedding problems and modular forms, Collectanea Math. 48 (1997), 63-83. MR 98j:11101
- 6.
- F. G. Frobenius, Über die Congruenz nach einem aus zwei endlichen Gruppen gebildeten Doppelmodul, J. Crelle 101 (1887), 279-299 (= Ges. Abh., II, 304-330).
- 7.
- F. G. Frobenius, Über Beziehungen zwischen den Primidealen eines algebraischen Körpers und den Substitutionen seiner Gruppe, Sitz. Akad. Wiss. Berlin (1896), 689-703 (= Ges. Abh., II, 719-733).
- 8.
- F. G. Frobenius, Über auflösbare Gruppen IV, Sitz. Akad. Wiss. Berlin (1901), 1216-1230 (= Ges. Abh., III, 189-203).
- 9.
- C. Jordan, Recherches sur les substitutions, J. Liouville 17 (1872), 351-367 (= Oe. I. 52).
- 10.
- L. Kronecker, Über die Irreductibilität von Gleichungen, Sitz. Akad. Wiss. Berlin (1880), 155-162 (= Werke, II, 83-93).
- 11.
- H. W. Lenstra, Jr., and P. Stevenhagen, Chebotarëv and his density theorem, Math. Intelligencer 18 (1996), 26-37. MR 97e:11144
- 12.
- J. Martinet, Petits discriminants des corps de nombres, Journées arithmétiques 1980 (J. V. Armitage, ed.), Cambridge Univ. Press, Cambridge, 1982, pp. 151-193. MR 84g:12009
- 13.
- R. W. K. Odoni, On the norms of algebraic integers, Mathematika 22 (1975), 71-80. MR 54:12715
- 14.
- M. Rubinstein and P. Sarnak, Chebyshev's Bias, Experiment. Math. 3 (1994), 173-197. MR 96d:11099
- 15.
- E. S. Selmer, On the irreducibility of certain trinomials, Math. Scand. 4 (1956), 287-302. MR 19:7f
- 16.
- J-P. Serre, Divisibilité de certaines fonctions arithmétiques, L'Ens. Math. 22 (1976), 227-260 (= Oe. 108). MR 55:7958
- 17.
- J-P. Serre, Modular forms of weight one and Galois representations, Algebraic Number Fields (A. Fröhlich, ed.), Acad. Press, London, 1977, pp. 193-268 (= Oe. 110). MR 56:8497
- 18.
- J-P. Serre, Quelques applications du théorème de densité de Chebotarev, Publ. Math. I.H.E.S. 54 (1981), 123-201 (= Oe. 125). MR 83k:12011
- 19.
- N. Tschebotareff (Chebotarev), Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse gehören, Math. Ann. 95 (1925), 191-228.
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Additional Information
Jean-Pierre Serre
Affiliation:
Collège de France, 3, Rue d’Ulm, Paris, France
Email:
serre@dmi.ens.fr
DOI:
http://dx.doi.org/10.1090/S0273-0979-03-00992-3
PII:
S 0273-0979(03)00992-3
Received by editor(s):
March 1, 2003
Posted:
July 17, 2003
Additional Notes:
This text first appeared in Math Medley 29 (2002), 3–18. The writing was done with the help of Heng Huat Chan. ©2002 Singapore Mathematical Society. Reprinted with permission
Article copyright:
© Copyright 2002 Singapore Mathematical Society
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