Commutator - Maple Help

GroupTheory

 Commutator
 construct the commutator of two subgroups
 DerivedSubgroup
 construct the derived subgroup of a group
 IsPerfect
 determine if a group is perfect

 Calling Sequence Commutator( A, B, G ) DerivedSubgroup( G ) IsPerfect( G )

Parameters

 G - a permutation group A - a permutation group B - a permutation group

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $A≔\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[1,2,3\right]\right]\right),\mathrm{Perm}\left(\left[\left[1,2\right]\right]\right)\right]\right):$
 > $B≔\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[2,3,4\right]\right]\right),\mathrm{Perm}\left(\left[\left[3,4\right]\right]\right)\right]\right):$
 > $C≔\mathrm{Commutator}\left(A,B,\mathrm{Symm}\left(4\right)\right)$
 ${C}{≔}\left[⟨\left({1}{,}{2}{,}{3}\right){,}\left({1}{,}{2}\right)⟩{,}⟨\left({2}{,}{3}{,}{4}\right){,}\left({3}{,}{4}\right)⟩\right]$ (1)
 > $\mathrm{GroupOrder}\left(C\right)$
 ${12}$ (2)
 > $C≔\mathrm{Commutator}\left(A,B,\mathrm{Symm}\left(5\right)\right)$
 ${C}{≔}\left[⟨\left({1}{,}{2}{,}{3}\right){,}\left({1}{,}{2}\right)⟩{,}⟨\left({2}{,}{3}{,}{4}\right){,}\left({3}{,}{4}\right)⟩\right]$ (3)
 > $\mathrm{GroupOrder}\left(C\right)$
 ${60}$ (4)
 > $G≔\mathrm{PermutationGroup}\left(\left\{\left[\left[1,2\right]\right],\left[\left[1,2,3\right],\left[4,5\right]\right]\right\}\right)$
 ${G}{≔}⟨\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}\right)⟩$ (5)
 > $\mathrm{DerivedSubgroup}\left(G\right)$
 $\left[⟨\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}\right)⟩{,}⟨\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}\right)⟩\right]$ (6)
 > $H≔\mathrm{DerivedSubgroup}\left(\mathrm{AlternatingGroup}\left(4\right)\right)$
 ${H}{≔}\left[{{\mathbf{A}}}_{{4}}{,}{{\mathbf{A}}}_{{4}}\right]$ (7)
 > $\mathrm{GroupOrder}\left(H\right)$
 ${4}$ (8)
 > $\mathrm{IsPerfect}\left(\mathrm{AlternatingGroup}\left(6\right)\right)$
 ${\mathrm{true}}$ (9)
 > $\mathrm{GroupOrder}\left(\mathrm{DerivedSubgroup}\left(\mathrm{AlternatingGroup}\left(6\right)\right)\right)$
 ${360}$ (10)
 > $\mathrm{IsPerfect}\left(\mathrm{PSL}\left(3,3\right)\right)$
 ${\mathrm{true}}$ (11)

The special linear group SL( 2, 5 ) is an example of a non-simple finite perfect group.

 > $\mathrm{IsPerfect}\left(\mathrm{SL}\left(2,5\right)\right)$
 ${\mathrm{true}}$ (12)
 > $\mathrm{IsSimple}\left(\mathrm{SL}\left(2,5\right)\right)$
 ${\mathrm{false}}$ (13)

Compatibility

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