§ 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 noexpression in terms of $[\alpha, \beta]$.

• 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.