MathieuGroup - Maple Help

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

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{MathieuGroup}\left(11\right)$
 ${{M}}_{{11}}$ (1)
 > $G≔\mathrm{MathieuGroup}\left(23\right)$
 ${G}{≔}{{M}}_{{23}}$ (2)
 > $\mathrm{type}\left(G,'\mathrm{PermutationGroup}'\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${10200960}$ (4)
 > $\mathrm{Transitivity}\left(G\right)$
 ${4}$ (5)
 > $G≔\mathrm{MathieuGroup}\left(12\right)$
 ${G}{≔}{{M}}_{{12}}$ (6)
 > $\mathrm{Degree}\left(G\right)$
 ${12}$ (7)
 > $\mathrm{Transitivity}\left(G\right)$
 ${5}$ (8)
 > $G≔\mathrm{MathieuGroup}\left(9\right):$
 > $\mathrm{IsSoluble}\left(G\right)$
 ${\mathrm{true}}$ (9)
 > $\mathrm{DerivedSeries}\left(G\right)$
 ${{M}}_{{9}}{▹}\left[{{M}}_{{9}}{,}{{M}}_{{9}}\right]{▹}\left[\left[{{M}}_{{9}}{,}{{M}}_{{9}}\right]{,}\left[{{M}}_{{9}}{,}{{M}}_{{9}}\right]\right]{▹}⟨⟩$ (10)
 > $\mathrm{Display}\left(\mathrm{CharacterTable}\left(\mathrm{MathieuGroup}\left(10\right)\right)\right)$

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

 > $G≔\mathrm{MathieuGroup}\left(11,'\mathrm{form}'="fpgroup"\right)$
 ${G}{≔}⟨{}{a}{,}{b}{}{\mid }{}{{a}}^{{2}}{,}{{b}}^{{4}}{,}{a}{}{{b}}^{{2}}{}{a}{}{{b}}^{{2}}{}{a}{}{{b}}^{{2}}{}{a}{}{{b}}^{{2}}{}{a}{}{{b}}^{{2}}{}{a}{}{{b}}^{{2}}{,}{a}{}{b}{}{a}{}{b}{}{a}{}{{b}}^{{-1}}{}{a}{}{b}{}{a}{}{{b}}^{{2}}{}{a}{}{{b}}^{{-1}}{}{a}{}{b}{}{a}{}{{b}}^{{-1}}{}{a}{}{{b}}^{{-1}}{,}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}⟩$ (11)

Compatibility

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