Corollary 2

Let G be a group. The quotient group G/D(G) is commutative. Let $\pi : G \to G/D(G)$ be the canonical homomorphism. Every homomorphism f of G into a commutative group G' can be expressed uniquely in the form $f = \bar{f} \circ \pi$, where $\bar{f} : G/D(G) \to G'$ is a homomorphism.

Now $\pi(D(G)) = \{e\}$. As $\pi$ is surjective, it follows that $D(G/D(G)) = \{e\}$, whence the first assertion. The second follows from § 4, no. 4, Proposition 5.