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On the Mathieu group M23

Abstract

Until 1965, when Janko [7] established the existence of his finite simple group J1, the five Mathieu groups were the only known examples of isolated finite simple groups. In 1951, R. G. Stanton [10] showed that M12 and M24 were determined uniquely by their order. Recent characterizations of M22 and M23 by Janko [8], M22 by D. Held [6], and M11 by W. J. Wong [12], have facilitated the unique determination of the three remaining Mathieu groups by their orders. D. Parrott [9] has so characterized M22 and M11, while this paper is an outline of the characterization of M23 in terms of its order.

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References
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[1]Brauer R., ‘Investigations on group characters’, Ann. of Math. 42 (1941), 936958.
[2]Brauer R., ‘On groups whose order contains a prime to the first power, I’, Amer. J. Math. 64 (1942), 401420.
[3]Brauer R., ‘Connection between ordinary and modular characters’, Ann. of Math. 42 (1941), 926935.
[4]Brauer R. and Tuan H. F., ‘On simple groups of finite orders, I’, Bull. Amer. Math. Soc. 51 (1945), 756766.
[5]Gorenstein D., Finite groups (Harper's Series in Modern Mathematics, Harper and Row, New York, 1968).
[6]Held D., ‘Eine Kennzeichnung der Mathieu-Gruppe M22 und der Alternierenden Gruppe A10’, J. of Alg. 8 (1968), 436449.
[7]Janko Z., ‘A new finite simple group with abelian Sylow 2-subgroups’, Proc. Nat. Acad. Sci. 53 (1965), 657658.
[8]Janko Z., ‘A characterization of the Mathieu Simple Groups I, II’, J. of Alg. 9 (1968), 1410.
[9]Parrott D., ‘On the Mathieu groups M22 and M11’, J. Aust. Math. Soc. 11 (1970), 6981.
[10]Stanton R. G., ‘The Mathieu Groups’, Canad. J. Math. 3 (1951), 164174.
[11]Wong S. K., ‘On a new finite non-abelian simple group of Janko’, to appear.
[12]Wong W. J., ‘On finite groups whose 2-Sylow subgroups have cyclic subgroups of index 2’, J. Aust. Math. Soc. 4 (1964), 90112.
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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