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Abstract Definitions for the Mathieu Groups M11and M12

Published online by Cambridge University Press:  20 November 2018

W.O.J. Moser
W.O.J. Moser*
Affiliation:
University of Saskatchewan
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A list of known finite simple groups has been given by Dickson [3, 4]. With but five exceptions, all of them fall into infinite families. The five exceptional groups, discovered by Mathieu [8,9], were further investigated by Jordan [7], Miller [10], de Séguier [11], Zassenhaus [13], and Witt [12]. In Witt's notation they are M11, M12, M22, M23, M24. Generators for them may be seen in the book of Carmichael [1, pp. 151, 263, 288]; but only for the smallest of them, M11 of order 7920, has a set of defining relations been given.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 2 , Issue 1 , 01 January 1959 , pp. 9 - 13
Copyright
Copyright © Canadian Mathematical Society 1959

References

1. Carmichael, R.D., Introduction to groups of finite order, (Boston, 1937).Google Scholar
2. Coxeter, H.S.M. and Moser, W.O.J., Generators and relations for discrete groups, Erg. d. Math N.F. Heft 14 (Berlin, 1957).Google Scholar
3. Dickson, L.E., Linear groups with an exposition of the Galois field theory, (Leipzig, 1901).Google Scholar
4. Dickson, L.E., A new system of simple groups, Math. Ann. 60 (1905), 137-150.Google Scholar
5. Frasch, H., Die Erzeugenden der Hauptkongruenzgruppen für Primzahlstufen, Math. Ann. 108 (1933), 229-252.Google Scholar
6. Fryer, K.D., A. class of permutation groups of prime degree, Canad. J. Math. 7 (1955), 24-34.Google Scholar
7. Jordan, C., Traité des substitutions, (Paris, 1870),Google Scholar
8. Mathieu, E., Mé moire sur 1'étude des fonctions de plusiers quantité s, J. de Math. (2) 6 (1861), 241-323.Google Scholar
9. Mathieu, E., Sur la fonction cinq fois transitive des 24 quantités, J. de Math. (2) 18 (1873), 25-46.Google Scholar
10. Miller, G. A., On the supposed five-fold transitive function of 24 elements, Messenger of Math. 27 (1898), 187-190.Google Scholar
11. de Séguier, J., Théorie des groupes finis, (Paris, 1904).Google Scholar
12. Witt, E., Die 5-fach transitiven Gruppen von Mathieu, Abh. Math. Sem. Univ. Hamburg 12 (1938), 256-264.Google Scholar
13. Zassenhaus, H., Über transitive Erweiterungen gewisser Gruppen aus Automorphismen endlicher mehrdimensionaler Geometrien, Math. Ann. 111 (1935), 748-759.Google Scholar