if $A$ and $B$ are subgroups of a group $G$, then their commutator $\left[A\,B\right]$ is the normal subgroup of $G$ generated by the commutators $\left[a\,b\right]$, for all elements $a$ in $A$ and $b$ in $B$.

The Commutator( A, B, G ) command computes the commutator of the subgroups A and B of G.

The derived subgroup (also called the commutator subgroup) of a group $G$ is the subgroup of $G$ generated by the commutators $\left[a\,b\right]$, as $a$ and $b$ range over the elements of $G$. Note that the derived subgroup of $G$ is the commutator $\left[G\,G\right]$. The quotient of $G$ by its derived subgroup is called the abelianization of $G$, and is the largest Abelian quotient of $G$.

The DerivedSubgroup( G ) command constructs the derived subgroup  of a group G. The group G must be an instance of a permutation group.

A group $G$ is said to be perfect if it is equal to its derived subgroup. For example, every non-Abelian simple group is perfect; however, there are perfect, but non-simple groups.

The IsPerfect( G ) command returns true if G is a perfect group, and returns false otherwise.

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

$A≔\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[1\,2\,3\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[1\,2\right]\right]\right)\right]\right)\:$

$B≔\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[2\,3\,4\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[3\,4\right]\right]\right)\right]\right)\:$

$C≔\mathrm{Commutator}\left(A\,B\,\mathrm{Symm}\left(4\right)\right)$

$C≔\left[⟨\left(1\,2\,3\right)\,\left(1\,2\right)⟩\,⟨\left(2\,3\,4\right)\,\left(3\,4\right)⟩\right]$

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

$12$

$C≔\mathrm{Commutator}\left(A\,B\,\mathrm{Symm}\left(5\right)\right)$

$C≔\left[⟨\left(1\,2\,3\right)\,\left(1\,2\right)⟩\,⟨\left(2\,3\,4\right)\,\left(3\,4\right)⟩\right]$

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

$60$

$G≔\mathrm{PermutationGroup}\left(\left\{\left[\left[1\,2\right]\right]\,\left[\left[1\,2\,3\right]\,\left[4\,5\right]\right]\right\}\right)$

$G≔⟨\left(1\,2\right)\,\left(1\,2\,3\right)\left(4\,5\right)⟩$

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

$\left[⟨\left(1\,2\right)\,\left(1\,2\,3\right)\left(4\,5\right)⟩\,⟨\left(1\,2\right)\,\left(1\,2\,3\right)\left(4\,5\right)⟩\right]$

$H≔\mathrm{DerivedSubgroup}\left(\mathrm{AlternatingGroup}\left(4\right)\right)$

$H≔\left[{\mathbf{A}}_4\,{\mathbf{A}}_4\right]$

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

$4$

$\mathrm{IsPerfect}\left(\mathrm{AlternatingGroup}\left(6\right)\right)$

$\mathrm{true}$

$\mathrm{GroupOrder}\left(\mathrm{DerivedSubgroup}\left(\mathrm{AlternatingGroup}\left(6\right)\right)\right)$

$360$

$\mathrm{IsPerfect}\left(\mathrm{PSL}\left(3\,3\right)\right)$

$\mathrm{true}$

$\mathrm{IsPerfect}\left(\mathrm{SL}\left(2\,5\right)\right)$

$\mathrm{true}$

$\mathrm{IsSimple}\left(\mathrm{SL}\left(2\,5\right)\right)$

$\mathrm{false}$

The GroupTheory[DerivedSubgroup] and GroupTheory[IsPerfect] commands were introduced in Maple 17.

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

The GroupTheory[Commutator] command was introduced in Maple 2019.

For more information on Maple 2019 changes, see Updates in Maple 2019.