Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-12-02T20:37:11.205Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Mathieu group M23

Published online by Cambridge University Press:  09 April 2009

N. Bryce
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 12 , Issue 4 , November 1971 , pp. 385 - 392
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Brauer, R., ‘Investigations on group characters’, Ann. of Math. 42 (1941), 936–958.CrossRefGoogle Scholar
[2]Brauer, R., ‘On groups whose order contains a prime to the first power, I’, Amer. J. Math. 64 (1942), 401–420.Google Scholar
[3]Brauer, R., ‘Connection between ordinary and modular characters’, Ann. of Math. 42 (1941), 926–935.CrossRefGoogle Scholar
[4]Brauer, R. and Tuan, H. F., ‘On simple groups of finite orders, I’, Bull. Amer. Math. Soc. 51 (1945), 756–766.CrossRefGoogle Scholar
[5]Gorenstein, D., Finite groups (Harper's Series in Modern Mathematics, Harper and Row, New York, 1968).Google Scholar
[6]Held, D., ‘Eine Kennzeichnung der Mathieu-Gruppe M22 und der Alternierenden Gruppe A10’, J. of Alg. 8 (1968), 436–449.CrossRefGoogle Scholar
[7]Janko, Z., ‘A new finite simple group with abelian Sylow 2-subgroups’, Proc. Nat. Acad. Sci. 53 (1965), 657–658.CrossRefGoogle Scholar
[8]Janko, Z., ‘A characterization of the Mathieu Simple Groups I, II’, J. of Alg. 9 (1968), 1–410.CrossRefGoogle Scholar
[9]Parrott, D., ‘On the Mathieu groups M22 and M11’, J. Aust. Math. Soc. 11 (1970), 69–81.Google Scholar
[10]Stanton, R. G., ‘The Mathieu Groups’, Canad. J. Math. 3 (1951), 164–174.CrossRefGoogle Scholar
[11]Wong, S. K., ‘On a new finite non-abelian simple group of Janko’, to appear.Google Scholar
[12]Wong, W. J., ‘On finite groups whose 2-Sylow subgroups have cyclic subgroups of index 2’, J. Aust. Math. Soc. 4 (1964), 90–112.Google Scholar