§ 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 noexpression in terms of .
- In general, the elements of the commutator subgroup will be productsof commutators.
- It measures the degree of non-abelian-ness of the group. isthe largest quotient of that is abelian. Alternatively, is the smallest normal subgroup we need to quotient by to get an abelianquotient. This quotienting is called abelianization.