§ The commutator subgroup
Define the commutator of g,h as [g,h]≡ghg−1h−1.
The subgroup generated by all commutators
in a group is called as the commutator subgroup. Sometimes denoted as
[G,G].
- We need to consider generation. Consider the free group on 4 lettersG=⟨a,b,c,d⟩. Now [a,b]⋅[c,d] has noexpression in terms of [α,β].
- In general, the elements of the commutator subgroup will be productsof commutators.
- It measures the degree of non-abelian-ness of the group. G/[G,G] isthe largest quotient of G that is abelian. Alternatively, [G,G]is the smallest normal subgroup we need to quotient by to get an abelianquotient. This quotienting is called abelianization.