SymmetricGroup - Maple Help

GroupTheory

 SymmetricGroup

 Calling Sequence SymmetricGroup( n ) SymmetricGroup( n, form = f ) Symm( n ) Symm( n, form = f )

Parameters

 n - algebraic ; understood to be a positive integer f - string ; either "permgroup" or "fpgroup"

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{SymmetricGroup}\left(7\right)$
 ${{\mathbf{S}}}_{{7}}$ (1)
 > $\mathrm{GroupOrder}\left(\mathrm{SymmetricGroup}\left(5\right)\right)$
 ${120}$ (2)
 > $\mathrm{IsPrimitive}\left(\mathrm{Symm}\left(11\right)\right)$
 ${\mathrm{true}}$ (3)
 > $G≔\mathrm{Symm}\left(4,'\mathrm{form}'="fpgroup"\right)$
 ${G}{≔}{{\mathbf{S}}}_{{4}}$ (4)
 > $G≔\mathrm{Symm}\left(n\right)$
 ${G}{≔}{{\mathbf{S}}}_{{n}}$ (5)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${n}{!}$ (6)
 > $\mathrm{IsSoluble}\left(G\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}n<5$
 ${\mathrm{true}}$ (7)

Compatibility

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