ElementOrder - Maple Help

GroupTheory

 ElementOrder
 compute the order of a group element
 ElementPower
 compute  powers of a group element

 Calling Sequence ElementOrder(g, G) ElementPower(g, n, G)

Parameters

 g - group element whose order is to be computed G - group containing the element g n - an integer

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Alt}\left(4\right)$
 ${G}{≔}{{\mathbf{A}}}_{{4}}$ (1)
 > $\mathrm{ElementOrder}\left(\mathrm{Perm}\left(\left[\left[1,2,4\right]\right]\right),G\right)$
 ${3}$ (2)
 > $\mathrm{PermOrder}\left(\mathrm{Perm}\left(\left[\left[1,2,4\right]\right]\right)\right)$
 ${3}$ (3)
 > $\mathrm{ElementPower}\left(\mathrm{Perm}\left(\left[\left[1,2,4\right]\right]\right),3,G\right)$
 $\left(\right)$ (4)
 > $\mathrm{ElementPower}\left(\mathrm{Perm}\left(\left[\left[1,2,4\right]\right]\right),2,G\right)$
 $\left({1}{,}{4}{,}{2}\right)$ (5)
 > $C≔\mathrm{CayleyTableGroup}\left(G\right)$
 ${C}{≔}{\mathrm{< a Cayley table group with 12 elements >}}$ (6)
 > $\mathrm{ElementOrder}\left(2,C\right)$
 ${2}$ (7)
 > $\mathrm{ElementOrder}\left(\mathrm{ElementPower}\left(2,3,C\right),C\right)$
 ${2}$ (8)

Compatibility

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