Example

(1) If G is a non-commutative simple group, then $D(G) = G$. Therefore every homomorphism of G into a commutative group is trivial.

(2) The derived group of the symmetric group $S_n$ is the alternating group $A_n$. For $A_n$ is generated by the products of two transpositions; if $\tau = \tau_{x,y}$ and $\tau' = \tau_{x',y'}$ are two transpositions, let $\sigma$ be a permutation such that $\sigma(x') = x$ and $\sigma(y') = y$. Then $\tau' = \sigma^{-1} \tau \sigma$ and $\tau \tau' = \tau^{-1} \tau' = \tau^{-1} \sigma^{-1} \tau \sigma$ is a commutator. Hence $A_n \subset D(S_n)$. As $S_n/A_n$ is commutative, $A_n \supset D(S_n)$ (Corollary 3).