FreeGroup - Maple Help

GroupTheory

 FreeGroup
 construct a free group of given rank or on a specified basis

 Calling Sequence FreeGroup( n ) FreeGroup( B )

Parameters

 n - nonnegint: the rank of the free group B - {set,list}(symbol) : a set or list of symbols specifying a basis

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{FreeGroup}\left(2\right)$
 $⟨{}{\mathrm{_x1}}{,}{\mathrm{_x2}}{}{\mid }{}{}⟩$ (1)
 > $F≔\mathrm{FreeGroup}\left(\left\{a,b,c\right\}\right)$
 ${F}{≔}⟨{}{a}{,}{b}{,}{c}{}{\mid }{}{}⟩$ (2)
 > $\mathrm{Generators}\left(F\right)$
 $\left[\left[{a}\right]{,}\left[{b}\right]{,}\left[{c}\right]\right]$ (3)
 > $\mathrm{GroupOrder}\left(F\right)$
 ${\mathrm{\infty }}$ (4)
 > $\mathrm{IsAbelian}\left(F\right)$
 ${\mathrm{false}}$ (5)
 > $\mathrm{IsAbelian}\left(\mathrm{FreeGroup}\left(1\right)\right)$
 ${\mathrm{true}}$ (6)
 > $\mathrm{latex}\left(\mathrm{FreeGroup}\left(2k\right)\right)$
 \mathrm{F}_{2 k}

Compatibility

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