Definition 4

Let $G$ be a group. The subgroup generated by the commutators of elements of $G$ is called the derived group of $G$.

The derived group of $G$ is thus the subgroup $(G, G)$. It is also denoted by $D(G)$. By an abuse of language, it is sometimes called the commutator group of $G$ although it is in general distinct from the set of commutators of elements of $G$ (Exercise 16). $D(G) = \{e\}$ if and only if $G$ is commutative.