An elementary Abelian group of order $p^n$, where $p$ is a prime number, is a direct product (or direct sum) of $n$ copies of the cyclic group of order $p$.

The ElementaryGroup( p, n ) command returns an elementary Abelian group of order $p^n$, either as a permutation group or as a finitely presented group, according to the value of the form option. By default, a permutation group is returned.

If either of p and n is not an explicit integer, then a symbolic group representing the elementary group of order $p^n$ is returned.

In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

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

$G≔\mathrm{ElementaryGroup}\left(3\,4\right)$

$G≔C_3^4$

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

$81$

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

$\mathrm{true}$

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

$\left[\left(1\,2\,3\right)\,\left(4\,5\,6\right)\,\left(7\,8\,9\right)\,\left(10\,11\,12\right)\right]$

$G≔\mathrm{ElementaryGroup}\left(3\,5\,'\mathrm{form}'=''fpgroup''\right)$

$G≔C_3^5$

$G≔\mathrm{ElementaryGroup}\left(17\,3\right)$

$G≔C_{17}^3$

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

$4913$

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

$p^{n+m}$

$\mathrm{IsCyclic}\left(\mathrm{ElementaryGroup}\left(p\&comma;n\right)\right)\mathbf{assuming}1<n$

$\mathrm{false}$

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

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

The GroupTheory[ElementaryGroup] command was updated in Maple 2020.