IsAbelian - Maple Help
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GroupTheory

 IsAbelian
 attempt to determine whether a group is Abelian (commutative)
 IsCommutative
 attempt to determine whether a group is commutative

 Calling Sequence IsAbelian( G ) IsCommutative( G )

Parameters

 G - a group

Description

 • A group $G$ is Abelian (or commutative) if every pair of elements of $G$ commute with each other.  That is, for all $a$ and $b$ in $G$, we have $a·b=b·a$.
 • A group is Abelian precisely when it is equal to its own center.
 • The IsAbelian( G ) command attempts to determine whether the group G is Abelian.  It returns true if G is Abelian and returns false otherwise. The command may raise an exception on (most) finitely presented groups, as shown in the last example below.
 • The IsCommutative command is provided as an alias.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{SmallGroup}\left(32,1\right):$
 > $\mathrm{IsAbelian}\left(G\right)$
 ${\mathrm{true}}$ (1)
 > $\mathrm{IsAbelian}\left(\mathrm{SmallGroup}\left(32,5\right)\right)$
 ${\mathrm{false}}$ (2)
 > $\mathrm{IsAbelian}\left(\mathrm{QuasicyclicGroup}\left(3\right)\right)$
 ${\mathrm{true}}$ (3)
 > $G≔⟨a|{a}^{6}=1⟩$
 ${G}{≔}⟨{}{a}{}{\mid }{}{{a}}^{{6}}{}⟩$ (4)
 > $\mathrm{IsAbelian}\left(G\right)$
 ${\mathrm{true}}$ (5)
 > $\mathrm{IsAbelian}\left(\mathrm{AGL}\left(2,8\right)\right)$
 ${\mathrm{false}}$ (6)
 > $\mathrm{IsCommutative}\left(⟨⟨a,b,c⟩|⟨\mathrm{.}\left(a,b\right)=\mathrm{.}\left(b,a\right),\mathrm{.}\left(a,c\right)=\mathrm{.}\left(c,a\right),\mathrm{.}\left(b,c\right)=\mathrm{.}\left(c,b\right)⟩⟩\right)$
 ${\mathrm{true}}$ (7)
 > $\mathrm{IsAbelian}\left(⟨⟨a,b⟩|⟨{a}^{2},{b}^{3},{\left(\mathrm{.}\left(a,b\right)\right)}^{5}=1⟩⟩\right)$

Compatibility

 • The GroupTheory[IsAbelian] and GroupTheory[IsCommutative] commands were introduced in Maple 17.
 • For more information on Maple 17 changes, see Updates in Maple 17.