Remark 2
Tag 06OK, permanent. Cite it as [Bourbaki, Tag 06OK].
Algebra, chapter V, § 11. Abelian extensions
The following result may be shown (cf. V, p. 165, Ex. 21): let G and H be two commutative groups all of whose elements are of finite order. Suppose that for every integer n ≥ 1, the equation nx = 0 has the same number of solutions, assumed finite in G and in H. Then the groups G and H are isomorphic. This provides a new proof of Prop. 2.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.