The order of an element $g$ of a group $G$ is the least positive integer $n$ such that $g^n$ is equal to the identity element of $G$, if one exists, and $\mathrm{\infty }$ otherwise.

The GroupOrder(g, G) command computes the order of the element g of the group G, if possible. Note that this is not always possible in case G is a finitely presented group.

Note that if g is a permutation, then ElementOrder(g, G) is equivalent to PermOrder(g).

The ElementPower( g, n, G ) command computes the power $g^n$ of the element g in the group G.

If g is a permutation, then ElementPower( g, n, G ) can be computed more simply as g^n.

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

$G≔\mathrm{Alt}\left(4\right)$

$G≔{\mathbf{A}}_4$

$\mathrm{ElementOrder}\left(\mathrm{Perm}\left(\left[\left[1\,2\,4\right]\right]\right)\,G\right)$

$3$

$\mathrm{PermOrder}\left(\mathrm{Perm}\left(\left[\left[1\,2\,4\right]\right]\right)\right)$

$3$

$\mathrm{ElementPower}\left(\mathrm{Perm}\left(\left[\left[1\,2\,4\right]\right]\right)\,3\,G\right)$

$\mathrm{ElementPower}\left(\mathrm{Perm}\left(\left[\left[1\,2\,4\right]\right]\right)\,2\,G\right)$

$\left(1\,4\,2\right)$

$C≔\mathrm{CayleyTableGroup}\left(⟨⟨⟨1\left|2\right|3\left|4\right|5|6⟩\,⟨2\left|1\right|4\left|3\right|6|5⟩\,⟨3\left|5\right|1\left|6\right|2|4⟩\,⟨4\left|6\right|2\left|5\right|1|3⟩\,⟨5\left|3\right|6\left|1\right|4|2⟩\,⟨6\left|4\right|5\left|2\right|3|1⟩⟩⟩\right)$

$C≔\mathrm{<\; a\; Cayley\; table\; group\; with\; 6\; elements\; >}$

$\mathrm{ElementOrder}\left(5\&comma;C\right)$

$3$

$\mathrm{ElementOrder}\left(\mathrm{ElementPower}\left(5\&comma;3\&comma;C\right)\&comma;C\right)$

$1$

The GroupTheory[ElementOrder] command was introduced in Maple 2015.

For more information on Maple 2015 changes, see Updates in Maple 2015.

The GroupTheory[ElementPower] command was introduced in Maple 2018.

For more information on Maple 2018 changes, see Updates in Maple 2018.