Corollary 4

Let G be a group and X a subset of G which generates G. The group D(G) is the normal subgroup of G generated by the commutators of elements of X.

Let H be the normal subgroup of G generated by the commutators of elements of X and $\phi : G \to G/H$ the canonical homomorphism. The set $\phi(X)$ generates $G/H$. The elements of $\phi(X)$ are pairwise permutable and hence H is commutative (§ 4, no. 3, Corollary 2 to Proposition 2). Hence (Corollary 3) H contains D(G). On the other hand, obviously $H \subset D(G)$.