The alternating group ${\mathbf{A}}_n$ on $n$ elements is the set of all even permutations of$\left\{1\&comma;2\&comma;\dots \&comma;n\right\}$ for a positive integer $n$. The order of ${\mathbf{A}}_n$ is equal to $\frac{n!}{2}$, for $1<n$. The alternating group of degree $n$ is simple if $n$ is at least $5$.

The AlternatingGroup( n ) command returns an alternating permutation group of degree n.  You can also use Alt( n ) as an abbreviation of AlternatingGroup( n ).

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".

If the argument n is not an integer constant, then a symbolic group is returned. In this case, the form option is ignored.

In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\&colon;$

$G≔\mathrm{AlternatingGroup}\left(7\right)$

$G≔{\mathbf{A}}_7$

$\mathrm{GroupOrder}\left(G\right)$

$2520$

$\mathrm{IsTransitive}\left(G\right)$

$\mathrm{true}$

$\mathrm{IsPrimitive}\left(G\right)$

$\mathrm{true}$

$\mathrm{IsSimple}\left(G\right)$

$\mathrm{true}$

$G≔\mathrm{Alt}\left(4\right)$

$G≔{\mathbf{A}}_4$

$\mathrm{IsSimple}\left(G\right)$

$\mathrm{false}$

$\mathrm{DrawSubgroupLattice}\left(G\right)$

$G≔\mathrm{Alt}\left(5\&comma;'\mathrm{form}'=''fpgroup''\right)$

$G≔⟨s\&comma;t\mid s^3\&comma;t^3\&comma;t{s}^{-1}tst{s}^{-1}ts\&comma;ststststst⟩$

$G≔\mathrm{Alt}\left(3n+7\right)$

$G≔{\mathbf{A}}_{3n+7}$

$\mathrm{IsSimple}\left(G\right)\mathbf{assuming}n::\mathrm{posint}$

$\mathrm{true}$

$\mathrm{GroupOrder}\left(G\right)$

$\frac{\left(3n+7\right)!}{2}$

The GroupTheory[AlternatingGroup] command was introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.