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The mathematical life of Cauchy’s group theorem. (English) Zbl 1065.01009

The first nontrivial result in permutation groups was the following theorem (stated but not proved by E. Galois, published 1846 by J. Liouville) of Cauchy (1844): Every group whose order is divisible by a prime number \(p\) has a subgroup of order \(p\). The paper tells the story of that theorem. It starts with a detailed analysis of the (uncomplete) proof by Cauchy and proceeds through its reworkings by H. Dedekind (1855–1858), G. Frobenius (1887), L. Sylow (1872), C. Jordan (1870) and G. A. Miller (1898) up to the most recent by J. H. MacKay (1959).

MSC:

01A55 History of mathematics in the 19th century
01A60 History of mathematics in the 20th century
20-03 History of group theory
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[1] Belhoste, B., Cauchy, un mathématicien légitimiste au XIXe siècle (1985), Belin: Belin Paris, (quotations from the English edition: Augustin-Louis Cauchy. A Biography. Springer-Verlag, New York, Berlin, 1991) · Zbl 0593.01006
[2] Birkeland, B., Ludvig Sylow’s “Lectures on Algebraic Equations and Substitutions, Christiana, 1862.”, Hist. Math., 23, 182-199 (1996) · Zbl 0858.01017
[3] Burnside, W., Theory of Groups of Finite Order (1897), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, UK · JFM 28.0118.03
[4] Cajori, F., A History of Mathematics (1919), Macmillan: Macmillan New York · JFM 47.0035.12
[5] Oeuvres complètes, vol. 1 (1932), Gauthier-Villars: Gauthier-Villars Paris, pp. 64-169
[6] Oeuvres complètes, vol. 13 (1932), Gauthier-Villars: Gauthier-Villars Paris, pp. 171-282 (citations are from the reprinted edition)
[7] Oeuvres Complètes, vol. 9, first series (1896), Gauthier-Villars: Gauthier-Villars Paris, p. 277; vol. 10, pp. 1-68 (citations are from the reprinted edition)
[8] Dahan Dalmedico, A., Les travaux de Cauchy sur les substitutions. Étude de son approche du concept de groupe, Arch. Hist. Exact Sci., 23, 279-319 (1980) · Zbl 0467.01010
[9] Dedekind, R., (Fricke, R.; Noether, E.; Ore, O., Gesammelte mathematische Werke, 3 vols (1932), Vieweg: Vieweg Braunschweig), Reprinted Chelsea, Bronx, NY, 1968
[10] Del Centina, A., The manuscript of Abel’s Parisian mémoire found in its entirety, Hist. Math., 29, 65-69 (2002) · Zbl 0996.01008
[11] (Dieudonné, J., Abrégé d’histoire des mathématiques, 2 vols (1978), Hermann: Hermann Paris) · Zbl 0383.01009
[12] Dyck, W., Gruppentheoretische Studien, Math. Ann., 20, 1-23 (1882) · JFM 14.0097.01
[13] Edwards, H. M., Postscript to “The background of Kummer”s proof….”, Arch. Hist. Exact Sci., 17, 381-394 (1977) · Zbl 0364.01004
[14] Fraleigh, J. B., A First Course in Abstract Algebra (1989), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0697.00001
[15] Reprinted in: Gesammelte Abhandlungen, Serre, J.-P. (Ed.), vol. 2. Springer-Verlag, Berlin, pp. 301-303; Reprinted in: Gesammelte Abhandlungen, Serre, J.-P. (Ed.), vol. 2. Springer-Verlag, Berlin, pp. 301-303 · JFM 18.0107.01
[16] Reprinted in: Gesammelte Abhandlungen, Serre, J.-P. (Ed.), vol. 2. Springer-Verlag, Berlin, pp. 304-330; Reprinted in: Gesammelte Abhandlungen, Serre, J.-P. (Ed.), vol. 2. Springer-Verlag, Berlin, pp. 304-330 · JFM 19.0136.01
[17] (Bourgne, R.; Azra, J. P., Ecrits et mémoires mathématiques (1962), Gauthier-Villars: Gauthier-Villars Paris), 43-71, Reprinted in:
[18] Grattan-Guinness, I., Christianity and mathematics: kinds of link, and the rare occurrences after 1750, Physis. Rivista Internazionale di Storia della Scienza, 37, 467-500 (2000)
[19] Hall, M., The Theory of Groups (1959), Macmillan: Macmillan New York
[20] Hawkins, T., The Berlin school of mathematics, (Mehrtens, H.; Bos, H.; Schneider, I., Social History of Nineteenth Century Mathematics (1981), Birkhäuser: Birkhäuser Boston), 233-245
[21] Hawkins, T., The origins of the theory of group characters, Arch. Hist. Exact Sci., 7, 142-170 (1971) · Zbl 0217.29903
[22] Herstein, I. N., Topics in Algebra (1964), Blaisdell: Blaisdell New York · Zbl 0122.01301
[23] Hungerford, T. W., Algebra (1974), Springer-Verlag: Springer-Verlag New York · Zbl 0293.12001
[24] Jordan, C., Traité des substitutions et des équations algébriques (1870), Gauthier-Villars: Gauthier-Villars Paris · JFM 02.0280.02
[25] Kiernan, B. M., The development of Galois theory from Lagrange to Artin, Arch. Hist. Exact Sci., 8, 40-154 (1971) · Zbl 0231.01003
[26] Knapp, A. W., Group representations and harmonic analysis from Euler to Langlands, Notices Am. Math. Soc., 43, 410-415 (1996), 537-549 · Zbl 1044.43501
[27] Kuhn, T. S., The Structure of Scientific Revolutions (1970), Univ. of Chicago Press: Univ. of Chicago Press Chicago
[28] (Bollobás, B., Littlewood’s Miscellany (1986), Cambridge Univ. Press: Cambridge Univ. Press New York/Cambridge), Reprinted as: · Zbl 0051.00101
[29] Lutzen, J., The mathematical correspondence between Julius Petersen and Ludvig Sylow, (Demidov, S.; etal., Amphora. Festschrift for Hans Wussing (1992), Birkhäuser: Birkhäuser Boston), 439-467 · Zbl 0788.01039
[30] Mac Lane, S.; Birkhoff, G., Algebra (1967), Macmillan: Macmillan New York
[31] Mazzotti, M., The geometers of God: mathematics and reaction in the Kingdom of Naples, Isis, 89, 674-701 (1998) · Zbl 1135.01307
[32] McKay, J. H., Another proof of Cauchy’s group theorem, Am. Math. Monthly, 66, 119 (1959) · Zbl 0082.02601
[33] Miller, G. A., On an extension of Sylow’s theorem, Bull. Amer. Math. Soc., 4, 323-327 (1898) · JFM 29.0109.02
[34] Miller, G. A.; Blichfeldt, H. F.; Dickson, L. E., Theory and Applications of Finite Groups (1916), Wiley: Wiley New York, reprinted Dover, New York, 1961 · Zbl 0098.25103
[35] Neumann, P. M., On the date of Cauchy’s contributions to the founding of the theory of groups, Bull. Austral. Math. Soc., 40, 293-302 (1989) · Zbl 0679.01005
[36] Nicholson, J., The development and understanding of the concept of quotient group, Hist. Math., 20, 68-88 (1993) · Zbl 0767.01022
[37] (George Pólya, Collected Papers, Rota, G.C. (Ed.), vol. 4 (1984), MIT Press: MIT Press Cambridge, MA), 308-419, Reprinted in: · JFM 63.0547.04
[38] Pourciau, B., Reading the master: Newton and the birth of celestial mechanics, Amer. Math. Monthly, 104, 1-19 (1997) · Zbl 0891.01021
[39] Read, R. C., Pólya’s theorem and its progeny, Math. Mag., 60, 275-282 (1987) · Zbl 0636.05005
[40] Roth, R., A history of Lagrange’s theorem on groups, Math. Mag., 74, 99-108 (2001) · Zbl 1004.01006
[41] Rothman, T., Genius and biographers: the fictionalization of Evariste Galois, Amer. Math. Monthly, 89, 84-106 (1982) · Zbl 0499.01017
[42] (Scharlau, W., Richard Dedekind 183l/1981: eine Würdigung zu seinem 150. Geburtstag (1981), Vieweg: Vieweg Braunschweig) · Zbl 0483.01007
[43] Scharlau, W., Die Entdeckung der Sylow-Sätze, Hist. Math., 15, 40-52 (1988) · Zbl 0637.01006
[44] Scholz, E., Die Entstehung der Galoistheorie, (Scholz, E., Geschichte der Algebra. Eine Einführung (1990), BI Wissenschaftsverlag: BI Wissenschaftsverlag Mannheim)
[45] Speiser, A., Die Theorie der Gruppen von endlicher Ordnung (1937), Julius Springer: Julius Springer Berlin · JFM 63.0059.01
[46] Sylow, L., Théorèmes sur les groupes des substitutions, Math. Ann., 5, 584-594 (1872) · JFM 04.0056.02
[47] Toti Rigatelli, L., 1989. La mente algebrica. Storia dello sviluppo della teoria di Galois nel XIX secolo, Bramante Editrice, [n.p.]; Toti Rigatelli, L., 1989. La mente algebrica. Storia dello sviluppo della teoria di Galois nel XIX secolo, Bramante Editrice, [n.p.]
[48] van der Waerden, B. L., A History of Algebra from al-Khwarizmi to Emmy Noether (1985), Springer-Verlag: Springer-Verlag New York · Zbl 0569.01001
[49] Waterhouse, W. C., The early proofs of Sylow’s theorem, Arch. Hist. Exact Sci., 21, 279-290 (1980) · Zbl 0436.01006
[50] Weber, H., Lehrbuch der Algebra, 3 vols (1894), F. Vieweg: F. Vieweg Braunschweig
[51] Wussing, H., Die Genesis des abstrakten Gruppenbegriffes, The Genesis of the Abstract Group Concept (1984), VEB Deutscher Verlag der Wissenschaften: VEB Deutscher Verlag der Wissenschaften Berlin: MIT Press: VEB Deutscher Verlag der Wissenschaften: VEB Deutscher Verlag der Wissenschaften Berlin: MIT Press Cambridge, MA, ) · Zbl 0199.29101
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