Let \(G\) be a group. For \(a, b\in G\), the commutator of \(a\) and \(b\) is \([a,b] \coloneqq aba^{-1}b^{-1}\). The commutator subgroup of \(G\) is the subgroup generated by all commutators: \[ [G, G] \coloneqq \gens{ aba^{-1} b^{-1} \st a, b\in G }. \]
Let \(G\) be a group. Then \([G,G]\) is a normal subgroup of \(G\) and \(G/[G,G]\) is abelian. If \(N\normal G\) and \(G/N\) is abelian, then \([G, G]\leq N\).
An element of \([G, G]\) is a product of commutators, and need not itself be a commutator.