§ The commutator subgroup

Define the commutator of g,hg, h as [g,h]≡ghg−1h−1[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][G, G].
  • We need to consider generation. Consider the free group on 4 letters G=⟨a,b,c,d⟩G = \langle a, b, c, d \rangle. Now [a,b]⋅[c,d][a, b] \cdot [c, d] has no expression in terms of [α,β][\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]G/[G, G] is the largest quotient of GG that is abelian. Alternatively, [G,G][G, G]is the smallest normal subgroup we need to quotient by to get an abelian quotient. This quotienting is called abelianization.