§ The commutator subgroup
Define the commutator of as .
The subgroup generated by all commutators
in a group is called as the commutator subgroup. Sometimes denoted as
- We need to consider generation. Consider the free group on 4 letters . Now has no expression in terms of .
- In general, the elements of the commutator subgroup will be products of commutators.
- It measures the degree of non-abelian-ness of the group. is the largest quotient of that is abelian. Alternatively, is the smallest normal subgroup we need to quotient by to get an abelian quotient. This quotienting is called abelianization.