Wolfram Language & System Documentation Center

MathieuGroupM24

MathieuGroupM24

MathieuGroupM24[]

represents the sporadic simple Mathieu group .

Details

By default, MathieuGroupM24[] is represented as a permutation group acting on points {1,…,24}.

Background & Context

MathieuGroupM24[] represents the Mathieu group , which is a group of order . It is one of the 26 sporadic simple groups of finite order. The default representation of MathieuGroupM24 is as a permutation group on the points having two generators.
The Mathieu group is the ninth smallest of the sporadic finite simple groups. It was discovered (along with the other four Mathieu groups MathieuGroupM11, MathieuGroupM12, MathieuGroupM22 and MathieuGroupM24) by mathematician Émile Léonard Mathieu in the late 1800s, making these groups tied for first in chronological order of discovery among sporadic groups. MathieuGroupM24 is 5-transitive in the sense that there exists at least one group element mapping any unique 5-tuple of elements of MathieuGroupM24 to any other unique 5-tuple therein. In addition to its permutation representation, can be defined in terms of generators and relations as and can be built from the action of the projective special linear group on the projective plane over the field with four elements. Along with the other sporadic simple groups, the Mathieu groups played a foundational role in the monumental (and complete) classification of finite simple groups.
The usual group theoretic functions may be applied to MathieuGroupM24[], including GroupOrder, GroupGenerators, GroupElements and so on. A number of precomputed properties of the Mathieu group are available via FiniteGroupData[{"Mathieu",24},"prop"].
MathieuGroupM24 is related to a number of other symbols. Along with MathieuGroupM11, MathieuGroupM12, MathieuGroupM22 and MathieuGroupM23, MathieuGroupM24 is one of five groups cumulatively referred to as the so-called "first generation" of sporadic finite simple groups. It is also one of 20 so-called "happy" sporadic groups, which all appear as a subquotient of the monster group.

Examples

open all close all

Basic Examples (1)

Order of the group :

Generators of a permutation representation of the group :

Properties & Relations (1)

The stabilizer of the last point coincides with the representation provided for the Mathieu group :

Top