Also called the commutator subgroup of a group is the subgroup generated by all commutator elements of that group. Should be used with the (group-theory) tag.
Given a group $G$, it's derived subgroup, also often called it's commutator subgroup is the subgroup generated by all elements of the form $xyx^{-1}y^{-1}$ in the group. This commutator element $xyx^{-1}y^{-1}$ is often written more simply as $[x,y]$, and subgroup is denoted commonly as $[G,G]$ or as $G'$ or as $G^{(1)}$.
The quotient $G/G'$ is the largest abelian quotient of $G$. We can see this by noting that by quotienting by the element $xyx^{-1}y^{-1}$ we're imposing the relation $xy = yx$ on $G$.
