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GroupTheory

 MathieuGroup

 Calling Sequence MathieuGroup(n)

Parameters

 n - an integer in { 9, 10, 11, 12, 21, 22, 23, 24 }

Description

 • The Mathieu groups ${M}_{n}$, for $n$ in $\left\{9,10,11,12,21,22,23,24\right\}$ are a family of transitive permutation groups studied by Émile Mathieu in the late nineteenth century.  The simple groups in the family are examples of highly transitive groups. The Mathieu group ${M}_{n}$ is simple for $n$ in $\left\{11,12,21,22,23,24\right\}$.
 • Note that the Mathieu group ${M}_{21}$ of order $20160$ is simple, but is not sporadic, being isomorphic to the group $PSL\left(3,4\right)$ .
 • The MathieuGroup( n ) command returns a permutation group isomorphic to the Mathieu group of degree n, where the degree n must be in { 9, 10, 11, 12, 21, 22, 23, 24 }. This is a sporadic finite simple group for n=11, 12, 22, 23, 24.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{MathieuGroup}\left(11\right)$
 ${{M}}_{{11}}$ (1)
 > $\mathrm{GroupOrder}\left(\mathrm{MathieuGroup}\left(23\right)\right)$
 ${10200960}$ (2)
 > $G≔\mathrm{MathieuGroup}\left(12\right)$
 ${G}{:=}{{M}}_{{12}}$ (3)
 > $\mathrm{Degree}\left(G\right)$
 ${12}$ (4)

Compatibility

 • The GroupTheory[MathieuGroup] command was introduced in Maple 17.