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Mathieu Groups


The five Mathieu groups M_(11), M_(12), M_(22), M_(23), and M_(24) were the first sporadic groups discovered, having been found in 1861 and 1873 by Mathieu. Frobenius showed that all the Mathieu groups are subgroups of M_(24).

The sporadic Mathieu groups are implemented in the Wolfram Language as MathieuGroupM11[], MathieuGroupM12[], MathieuGroupM22[], MathieuGroupM23[], and MathieuGroupM24[].

All the sporadic Mathieu groups are multiply transitive. The following table summarizes some properties of the Mathieu groups, where k indicates the transitivity and L is the length of the minimal permutation support (from which the groups derive their designations).

groupkLorderfactorization
M_(11)41179202^4·3^2·5·11
M_(12)512950402^6·3^3·5·11
M_(22)3224435202^7·3^2·5·7·11
M_(23)423102009602^7·3^2·5·7·11·23
M_(24)5242448230402^(10)·3^3·5·7·11·23

The Mathieu groups are most simply defined as automorphism groups of Steiner systems, as summarized in the following table.

Mathieu groupSteiner system
M_(11)S(4,5,11)
M_(12)S(5,6,12)
M_(22)S(3,6,22)
M_(23)S(4,7,23)
M_(24)S(5,8,24)

See also

Automorphism Group, Large Witt Graph, Simple Group, Sporadic Group, Steiner System, Transitive Group, Witt Geometry

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References

Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, 1985.Conway, J. H. and Sloane, N. J. A. "The Golay Codes and the Mathieu Groups." Ch. 11 in Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, pp. 299-330, 1993.Dixon, J. and Mortimer, B. Permutation Groups. New York: Springer-Verlag, 1996.Rotman, J. J. Ch. 9 in An Introduction to the Theory of Groups, 4th ed. New York: Springer-Verlag, 1995.Wilson, R. A. "ATLAS of Finite Group Representation." http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/.

Referenced on Wolfram|Alpha

Mathieu Groups

Cite this as:

Weisstein, Eric W. "Mathieu Groups." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MathieuGroups.html

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