Mathieu group

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

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.

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

Revised on April 15, 2013 07:41:42
by David Roberts
(192.43.227.18)