 Commutator subgroup

The commutator subgroup $G^{\prime }$ is generated by all $[x,y]=x^{-1}{y}^{-1}xy$. It is characteristic, $G/G^{\prime }$ is abelian, and every normal subgroup with abelian quotient contains $G^{\prime }$.

The commutator subgroup of a group \( G \), often denoted as \( [G, G] \) or sometimes as \( G' \), is a specific subgroup of \( G \) that captures the "non-abelian" structure of the group. It is constructed using the commutators of elements from \( G \).