A free group is a group that has a free basis, which is a set $B$ for which the group has the presentation with $B$ as generators and an empty set of relators. The number of elements in a basis $B$ is called the rank of the free group.

The FreeGroup( n ) command returns a free group, as a finitely presented group, of rank $n$.

The FreeGroup( B ) command returns a free group with the member of the set or list $B$ of names as basis. Its rank is therefore the number of elements in $B$.

Note that a free group of rank $0$ is trivial, and a free group of rank $1$ is an infinite cyclic group. Free groups with rank greater than $1$ are non-abelian.

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

$\mathrm{FreeGroup}\left(2\right)$

$⟨\mathrm{\_x1}\,\mathrm{\_x2}\mid ⟩$

$F≔\mathrm{FreeGroup}\left(\left\{a\,b\,c\right\}\right)$

$F≔⟨a\,b\,c\mid ⟩$

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

$\left[\left[a\right]\,\left[b\right]\,\left[c\right]\right]$

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

$\mathrm{\infty }$

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

$\mathrm{false}$

$\mathrm{IsAbelian}\left(\mathrm{FreeGroup}\left(1\right)\right)$

$\mathrm{true}$

$\mathrm{latex}\left(\mathrm{FreeGroup}\left(2k\right)\right)$

\mathrm{F}_{2 k}

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

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