Proposition 6

Tag 05D2, permanent. Cite it as [Bourbaki, Tag 05D2].

Algebra, chapter I, § 6. Extensions, solvable groups, nilpotent groups

Let $f : G \to G'$ be a group homomorphism. Then $f(D(G)) \subset D(G')$ . If $f$ is surjective, the homomorphism of $D(G)$ into $D(G')$ the restriction of $f$ is surjective.

The image under $f$ of a commutator of elements of $G$ is a commutator of elements of $G'$ . If $f$ is surjective, the image under $f$ of the set of commutators of $G$ is the set of commutators of $G'$ . The proposition thus follows from § 4, no. 3, Corollary 3 to Proposition 2.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Proposition 12, § 6
Lemma 1, Appendix 1

Cites

§ 4, no. 3, Corollary 3 to Proposition 2