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.

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

$G≔\mathrm{SymmetricGroup}\left(5\right)$

$G≔{\mathbf{S}}_5$

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

$120$

$G≔\mathrm{BabyMonster}\left(\right)$

$G≔𝔹$

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

$4154781481226426191177580544000000$

$\mathrm{GroupOrder}\left(\mathrm{DihedralGroup}\left(3k+1\right)\right)$

$6k+2$

$\mathrm{GroupOrder}\left(\mathrm{ElementaryGroup}\left(p\,n\right)\right)$

$p^n$

$G≔⟨⟨a\,b⟩|⟨a^2\,b^3\,{\left(a·b\right)}^5=1⟩⟩$

$G≔⟨a\,b\mid a^2\,b^3\,ababababab⟩$

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

$60$

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

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