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The Mathieu Groups

Published online by Cambridge University Press:  20 November 2018

R. G. Stanton
R. G. Stanton*
Affiliation:
The University of Toronto
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An enumeration of known simple groups has been given by Dickson [17]; to this list, he made certain additions in later papers [15], [16]. However, with but five exceptions, all known simple groups fall into infinite families; these five unusual simple groups were discovered by Mathieu [21], [22] and, after occasioning some discussion [20], [23], [27], were relegated to the position, which they still hold, of freakish groups without known relatives. Further interest is attached to these Mathieu groups in virtue of their providing the only known examples (other than the trivial examples of the symmetric and alternating groups) of quadruply and quintuply transitive permutation groups.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 3 , 1951 , pp. 164 - 174
Copyright
Copyright © Canadian Mathematical Society 1951

References

[1] Brauer, R., On the Cartan Invariants of Groups of Finite Order, Ann. of Math., vol. 42 (1941), 53–61.Google Scholar
[2] Brauer, R., On the Connection Between the Ordinary and the Modular Characters of Groups of Finite Order, Ann. of Math., vol. 42 (1941), 926–935.Google Scholar
[3] Brauer, R., Investigations on Group Characters, Ann. of Math., vol. 42 (1941), 936–958.Google Scholar
[4] Brauer, R., On Groups whose Order Contains a Prime Number to the First Power, Am. J. of Math., vol. 64 (1942), 401–440.Google Scholar
[5] Brauer, R., On Permutation Groups of Prime Degree and Related Classes of Groups, Ann. of Math., vol. 44 (1943), 57–79.Google Scholar
[6] Brauer, R., On the Arithmetic in a Group Ring, Proc. Nat. Acad. Sciences, vol. 30 (1944), 109–114.Google Scholar
[7] Brauer, R., On Blocks of Characters of Groups of Finite Order, Proc. Nat. Acad. Sciences, vol. 32 (1946), 182-186 and 215–219.Google Scholar
[8] Brauer, R., On Modular and p-adic Representations of Algebras, Proc. Nat. Acad. Sciences, vol. 25 (1939), 252–258.Google Scholar
[9] Brauer, R., On ifo Representation of a Group of Order g in the Field of the g-th Roots of Unity, Am. J. of Math., vol. 67 (1945), 461–471.Google Scholar
[10] Brauer, R. and Nesbitt, C., On the Modular Representations of Groups of Finite Order, Univ. of Toronto Studies, No. 4 (1937).Google Scholar
[11] Brauer, R., On the Modular Characters of Groups, Ann. of Math., vol. 42 (1941), 556–590.Google Scholar
[12] Burnside, W., The Theory of Groups of Finite Order, Cambridge (1911).Google Scholar
[13] Carmichael, R., An Introduction to the Theory of Groups of Finite Order, Boston (1937).Google Scholar
[14] Dickson, L, On the Group Defined for any Given Field by the Multiplication Table of any Finite Group, T.A.M.S., vol. 3 (1902), 285–301.Google Scholar
[15] Dickson, L, Theory of Linear Groups in an Arbitrary Field, T.A.M.S., vol. 2 (1901), 363–394.Google Scholar
[16] Dickson, L, A new System of Simple Groups, Math. Ann., vol. 60 (1905), 137–150.Google Scholar
[17] Dickson, L, Linear Groups, Leipzig (1901).Google Scholar
[18] Frobenius, G., Über die Charactere der Symmetrischen Gruppe, Sitz. Preuss. Akad. Wissen. (1900), 516–534.Google Scholar
[19] Frobenius, G., Über die Charactere der Mehrfach transitiven Gruppen, Sitz. Preuss. Akad. Wissen. (1904), 558–571.Google Scholar
[20] Jordan, C., Traité des Substitutionst, Paris (1870).Google Scholar
[21] Mathieu, E., Mémoire sur Vétude des fonctions de plusieurs quantités, Jour, de Math., 2me Série, vol. 6 (1861), 241–323.Google Scholar
[22] Mathieu, E., Sur la fonction cinq fois transitive de 24 quantités, Jour, de Math., 2me Série, vol. 18 (1873), 25–46.Google Scholar
[23] Miller, G., On the Supposed Five-fold Transitive Function of 24 Elements, Mess, of Math., vol. 27 (1898), 187–190.Google Scholar
[24] Moore, E., Tactical Memoranda, Am. J. of Math., vol. 18 (1896), 268–275.Google Scholar
[25] Netto, E., Lehrbuch der Kombinatorik, Leipzig (1927).Google Scholar
[26] Schur, I., Neue Begründung der Théorie der Gruppencharactere, Sitz. Preuss. Akad. Wissen. (1905), 406–432.Google Scholar
[27] de Séguier, J., Théorie des Groupes Finis, Paris (1904).Google Scholar
[28] Speiser, A., Die Théorie der Gruppen, New York (1945).Google Scholar
[29] Tuan, H., On Groups whose Orders Contain a Prime to the First Power, Ann. of Math., vol. 45 (1944), 110–140.Google Scholar
[30] Witt, E., Die 5-fach Transitiven Gruppen von Mathieu, Abhand. Math. Sem. Hamb., Band 12 (1938), 256–264.Google Scholar
[31] Witt, E., Über Steinersche Systeme, Abhand. Math. Sem. Hamb., Band 12 (1938), 265–275.Google Scholar