GroupOrder - Maple Help

GroupTheory

 GroupOrder
 compute the order of a group

 Calling Sequence GroupOrder( G )

Parameters

 G - a permutation group

Description

 • The order of a group is the cardinality of its underlying set.
 • The GroupOrder( G ) command computes the order of the group G, if possible. (Note that the order of a finitely presented group cannot be determined, in general.)
 • In most cases, it is much more efficient to use the GroupOrder command to determine the order of a group than to list its elements with the Elements command and then count them.  In the case of a symbolic group, this is the only way to compute the group order.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{SymmetricGroup}\left(5\right)$
 ${G}{≔}{{\mathbf{S}}}_{{5}}$ (1)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${120}$ (2)
 > $G≔\mathrm{BabyMonster}\left(\right)$
 ${G}{≔}{𝔹}$ (3)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${4154781481226426191177580544000000}$ (4)
 > $\mathrm{GroupOrder}\left(\mathrm{DihedralGroup}\left(3k+1\right)\right)$
 ${6}{}{k}{+}{2}$ (5)
 > $\mathrm{GroupOrder}\left(\mathrm{ElementaryGroup}\left(p,n\right)\right)$
 ${{p}}^{{n}}$ (6)
 > $G≔⟨⟨a,b⟩|⟨{a}^{2},{b}^{3},{\left(a·b\right)}^{5}=1⟩⟩$
 ${G}{≔}⟨{}{a}{,}{b}{}{\mid }{}{{a}}^{{2}}{,}{{b}}^{{3}}{,}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}⟩$ (7)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${60}$ (8)

Compatibility

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