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GroupTheory

 MathieuGroup

 Calling Sequence MathieuGroup( n, formopt )

Parameters

 n - an integer in { 9, 10, 11, 12, 21, 22, 23, 24 } formopt - (optional) equation of the form form = F, where F is either "permgroup" (the default) or "fpgroup"

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 while the Mathieu group ${M}_{21}$ of order $20160$ is simple, it 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.
 • The Mathieu group ${M}_{9}$ is, in fact a soluble group.
 • The form = F option controls the form of the group returned. By default, a permutation group is returned; this is equivalent to passing the option form = "permgroup". A finitely presented group can be obtained by passing the option form = "fpgroup".
 • In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

Examples

 > with(GroupTheory):
 > MathieuGroup( 11 );
 ${\mathrm{GroupTheory}}{:-}{\mathrm{MathieuGroup}}{}\left({11}\right)$ (1)
 > G := MathieuGroup( 23 );
 ${\mathrm{GroupTheory}}{:-}{\mathrm{MathieuGroup}}{}\left({23}\right)$ (2)
 > type( G, 'PermutationGroup' );
 ${\mathrm{true}}$ (3)
 > GroupOrder( G );
 ${10200960}$ (4)
 > Transitivity( G );
 ${4}$ (5)
 > G := MathieuGroup( 12 );
 ${\mathrm{GroupTheory}}{:-}{\mathrm{MathieuGroup}}{}\left({12}\right)$ (6)
 > Degree( G );
 ${12}$ (7)
 > Transitivity( G );
 ${5}$ (8)
 > G := MathieuGroup( 9 ):
 > IsSoluble( G );
 ${\mathrm{true}}$ (9)
 > DerivedSeries( G );
 ${\mathrm{GroupTheory}}{:-}{\mathrm{DerivedSeries}}{}\left({\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}\right)$ (10)
 > Display( CharacterTable( MathieuGroup( 10 ) ) );

 C 1a 2a 3a 4a 4b 5a 8a 8b |C| 1 45 80 90 180 144 90 90 $\mathrm{χ__1}$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $\mathrm{χ__2}$ $1$ $1$ $1$ $1$ $-1$ $1$ $-1$ $-1$ $\mathrm{χ__3}$ $9$ $1$ $0$ $1$ $-1$ $-1$ $1$ $1$ $\mathrm{χ__4}$ $9$ $1$ $0$ $1$ $1$ $-1$ $-1$ $-1$ $\mathrm{χ__5}$ $10$ $-2$ $1$ $0$ $0$ $0$ $-I\sqrt{2}$ $I\sqrt{2}$ $\mathrm{χ__6}$ $10$ $-2$ $1$ $0$ $0$ $0$ $I\sqrt{2}$ $-I\sqrt{2}$ $\mathrm{χ__7}$ $10$ $2$ $1$ $-2$ $0$ $0$ $0$ $0$ $\mathrm{χ__8}$ $16$ $0$ $-2$ $0$ $0$ $1$ $0$ $0$

 > G := MathieuGroup( 11, 'form' = "fpgroup" );
 ${\mathrm{GroupTheory}}{:-}{\mathrm{FPGroup}}{}\left(\left[{a}{,}{b}\right]{,}\left\{\left[{a}{,}{a}\right]{,}\left[{b}{,}{b}{,}{b}{,}{b}\right]{,}\left[{a}{,}{b}{,}{b}{,}{a}{,}{b}{,}{b}{,}{a}{,}{b}{,}{b}{,}{a}{,}{b}{,}{b}{,}{a}{,}{b}{,}{b}{,}{a}{,}{b}{,}{b}\right]{,}\left[{a}{,}{b}{,}{a}{,}{b}{,}{a}{,}\frac{{1}}{{b}}{,}{a}{,}{b}{,}{a}{,}{b}{,}{b}{,}{a}{,}\frac{{1}}{{b}}{,}{a}{,}{b}{,}{a}{,}\frac{{1}}{{b}}{,}{a}{,}\frac{{1}}{{b}}\right]{,}\left[{a}{,}{b}{,}{a}{,}{b}{,}{a}{,}{b}{,}{a}{,}{b}{,}{a}{,}{b}{,}{a}{,}{b}{,}{a}{,}{b}{,}{a}{,}{b}{,}{a}{,}{b}{,}{a}{,}{b}{,}{a}{,}{b}\right]\right\}\right)$ (11)
 >

Compatibility

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