nLab Mathieu group



Group Theory

Exceptional structures



The Mathieu groups, denoted M 11M_{11}, M 12M_{12}, M 22M_{22}, M 23M_{23} and M 24M_{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 11M_{11} - 7920;
  • M 12M_{12} - 95040;
  • M 22M_{22} - 443520;
  • M 23M_{23} - 10200960;
  • M 24M_{24} - 244823040;

The Matthieu group M 24M_{24} is the automorphism group of the binary Golay code; this is a vector space over the field 𝔽 2\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 24M_{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 13M_{13}.

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

See also

Last revised on August 27, 2019 at 12:13:54. See the history of this page for a list of all contributions to it.