§ The commutator subgroup

Define the commutator of $g, h$ as $[g, h] \equiv ghg^{-1}h^{-1}$. 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 letters $G = \langle a, b, c, d \rangle$. Now $[a, b] \cdot [c, d]$ has no expression in terms of $[\alpha, \beta]$.

• In general, the elements of the commutator subgroup will be products of commutators.

• It measures the degree of non-abelian-ness of the group. $G/[G, G]$ is the largest quotient of $G$ that is abelian. Alternatively, $[G, G]$is the smallest normal subgroup we need to quotient by to get an abelian quotient. This quotienting is called abelianization.