The symmetric group ${\mathbf{S}}_n$ on $n$ elements is the set of all permutations of$\left\{1\,2\,\dots \,n\right\}$ for a positive integer $n$.

The SymmetricGroup( n ) command returns a symmetric permutation group of degree n.  The command Symm( n ) is also available as an abbreviation for SymmetricGroup( n ).

By default, a permutation group is returned.  However, you can request that a finitely presented group isomorphic to the symmetric group of the indicated degree be returned by using the 'form' = "fpgroup" option. The default for the form option is 'form' = "permgroup".

If the degree n is not a positive integer, then a symbolic group representing the symmetric group of degree n is returned.

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

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$\mathrm{SymmetricGroup}\left(7\right)$

${\mathbf{S}}_7$

$\mathrm{GroupOrder}\left(\mathrm{SymmetricGroup}\left(5\right)\right)$

$120$

$\mathrm{IsPrimitive}\left(\mathrm{Symm}\left(11\right)\right)$

$\mathrm{true}$

$G≔\mathrm{Symm}\left(4\,'\mathrm{form}'=''fpgroup''\right)$

$G≔{\mathbf{S}}_4$

$G≔\mathrm{Symm}\left(n\right)$

$G≔{\mathbf{S}}_n$

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

$n!$

$\mathrm{IsSoluble}\left(G\right)\mathbf{assuming}n<5$

$\mathrm{true}$

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

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

The GroupTheory[SymmetricGroup] command was updated in Maple 2021.