Remark

(1) Corollary 2 can also be expressed by saying that $G/D(G)$, together with $\pi$, is a solution of the universal mapping problem for G, relative to commutative groups and homomorphisms from G to commutative groups.

(2) Under the hypotheses of Corollary 4, the subgroup generated by the commutators of elements of X is contained in D(G) but is not in general equal to D(G) (cf. Exercise 15e).