Let G be a group and let G′=⟨aba−1b−1⟩; that is, G′ is the subgroup of all finite products of elements in G of the form aba−1b−1. The subgroup G′ is called the commutator subgroup of G.
(a) Show that G′ is a normal subgroup of G.
(b) Let N be a normal subgroup of G. Prove that G/N is abelian if and only if N contains the commutator subgroup of G.
Hint. (a) Let g∈G and h∈G′. If h=aba−1b−1, then
ghg−1=gaba−1b−1g−1=(gag−1)(gbg−1)(ga−1g−1)(gb−1g−1)=(gag−1)(gbg−1)(gag−1)−1(gbg−1)−1.
We also need to show that if h=h1⋯hn with hi=aibiai−1bi−1, then ghg−1 is a product of elements of the same type. However, ghg−1=gh1⋯hng−1=(gh1g−1)(gh2g−1)⋯(ghng−1).