nLab Mathieu group

Idea

The Mathieu groups, denoted $M_{11}$, $M_{12}$, $M_{22}$, $M_{23}$ and $M_{24}$ are five of the sporadic finite simple groups in the Happy Family. They were first described in the 1860-70s by Émile Mathieu, and the first such groups to be discovered.

They arise as the automorphism groups of Steiner systems. The orders of the groups are as follows:

• $M_{11}$ - 7920;
• $M_{12}$ - 95040;
• $M_{22}$ - 443520;
• $M_{23}$ - 10200960;
• $M_{24}$ - 244823040;

The Matthieu group $M_{24}$ is the automorphism group of the binary Golay code; this is a vector space over the field $\mathbb{F}_2$. The other groups can be obtained as stabilisers of various (sets of) elements of the Golay code, and hence are subgroups of $M_{24}$. The Mathieu groups form the so-called first generation of the happy family: the collection of 20 sporadic groups which are subgroups of the Monster group.

Related concepts

• Mathieu moonshine

• Fischer group

References

• N-cafe blogpost on the groupoid $M_{13}$.

• John Conway, Noam D. Elkies and Jeremy L. Martin, “The Mathieu group $M_{12}$ and its pseudogroup extension $M_{13}$”, Experimental Mathematics 15 (2006), 223–236. Eprint.

See also:

• Wikipedia, Mathieu group

On the rational homotopy of K3-surfaces in relation to the Mathieu group:

• Federico Carta, John F. R. Duncan, Yang-Hui He: Rational K3 Homotopy and the Largest Mathieu Group [arXiv:2509.18969]