GroupTheory
FreeGroup
construct a free group of given rank or on a specified basis
Calling Sequence
Parameters
Description
Examples
Compatibility
FreeGroup( n )
FreeGroup( B )
n

nonnegint: the rank of the free group
B
{set,list}(symbol) : a set or list of symbols specifying a basis
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 nonabelian.
$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$
$\mathrm{FreeGroup}\left(2\right)$
$\u27e8{}{\mathrm{\_x1}}{\,}{\mathrm{\_x2}}{}{\mid}{}{}\u27e9$
$F\u2254\mathrm{FreeGroup}\left(\left\{a\,b\,c\right\}\right)$
${F}{\u2254}\u27e8{}{a}{\,}{b}{\,}{c}{}{\mid}{}{}\u27e9$
$\mathrm{Generators}\left(F\right)$
$\left[\left[{a}\right]{\,}\left[{b}\right]{\,}\left[{c}\right]\right]$
$\mathrm{GroupOrder}\left(F\right)$
${\mathrm{\infty}}$
$\mathrm{IsAbelian}\left(F\right)$
${\mathrm{false}}$
$\mathrm{IsAbelian}\left(\mathrm{FreeGroup}\left(1\right)\right)$
${\mathrm{true}}$
$\mathrm{latex}\left(\mathrm{FreeGroup}\left(2k\right)\right)$
\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.
See Also
GroupTheory[FPGroup]
GroupTheory[Generators]
GroupTheory[GroupOrder]
GroupTheory[IsAbelian]
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