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Further elementary techniques using the miracle octad generator

Published online by Cambridge University Press:  20 January 2009

R. T. Curtis
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In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an “octad generator”; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 32 , Issue 3 , October 1989 , pp. 345 - 353
Copyright
Copyright © Edinburgh Mathematical Society 1989

References

REFERENCES

1.Benson, D. J., The Construction of J4 (Ph.D. thesis, Cambridge, 1980).Google Scholar
2.Conway, J. H., Three lectures on exceptional groups, in Finite Simple Groups (eds. Powell, M. B. and Higman, G., Academic Press, London, Orlando and New York), 215–247.Google Scholar
3.Conway, J. H., The Miracle Octad Generator, in Topics in Group Theory and Computation (Ed. Curran, M. P. J., Academic Press, London, Orlando and New York, 1977), 62–68.Google Scholar
4.Conway, J. H., Hexacode and tetracode—MOG and MINIMOG, in Computational Group Theory (Ed. Atkinson, Michael D., Academic Press, London, Orlando and New York, 1984), 359–365.Google Scholar
5.Curtis, R. T., On subgroups of 0, I: lattice stabilizers, J. Algebra 27 (1973), 549–573.CrossRefGoogle Scholar
6.Curtis, R. T., A new combinatorial approach to M 24, Math. Proc. Cambridge Philos. Soc. 79 (1976), 25–42.CrossRefGoogle Scholar
7.Curtis, R. T., On subgroups of 0, II local structure, J. Algebra 63 (1980), 413–434.CrossRefGoogle Scholar
8.Curtis, R. T., Eight octads suffice, J. Combin. Theory Ser. A 36 (1984), 116–123.CrossRefGoogle Scholar
9.Todd, J. A., A representation of the Mathieu group M 24 as a collineation group, Anc. Mat. Para Appl. Ser. IV 71 (1966), 199–238.CrossRefGoogle Scholar
10.Wilson, R. A., The maximal subgroups of Conway's group 2, J. Algebra 84 (1983), 107–114.CrossRefGoogle Scholar