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.

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

$G≔\mathrm{SmallGroup}\left(32\,1\right)\:$

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

$\mathrm{true}$

$\mathrm{IsAbelian}\left(\mathrm{SmallGroup}\left(32\,5\right)\right)$

$\mathrm{false}$

$\mathrm{IsAbelian}\left(\mathrm{QuasicyclicGroup}\left(3\right)\right)$

$\mathrm{true}$

$G≔⟨a|a^6=1⟩$

$G≔⟨a\mid a^6⟩$

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

$\mathrm{true}$

$\mathrm{IsAbelian}\left(\mathrm{AGL}\left(2\,8\right)\right)$

$\mathrm{false}$

$\mathrm{IsCommutative}\left(⟨⟨a\,b\,c⟩|⟨a·b=b·a\,a·c=c·a\,b·c=c·b⟩⟩\right)$

$\mathrm{true}$

$\mathrm{IsAbelian}\left(⟨⟨a\,b⟩|⟨a^2\,b^3\,{\left(a·b\right)}^5=1⟩⟩\right)$

Error, (in IsAbelian) cannot determine whether a general finitely presented group is Abelian.  If you know that your group is finite, try converting it to a permutation group by using the `PermutationGroup' command with your finitely presented group as input.

The GroupTheory[IsAbelian] and GroupTheory[IsCommutative] commands were introduced in Maple 17.

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