Corollary 7

Let G be a locally compact commutative group.

a) If G is divisible, then the dual group $\widehat{G}$ is torsion-free ;

b) If the dual group $\widehat{G}$ is torsion-free, and if $k\in \mathbf{Z}$ is nonzero, then the image of the homomorphism $x\mapsto x^k$ of G into G is dense in G ;

c) Suppose G discrete or compact. For G to be divisible it is necessary and sufficient that $\widehat{G}$ be torsion-free.

Assertions a) and b) follow from Cor. 6. If G is discrete or compact, the image of the morphism $x\mapsto x^k$ of G into G is closed, and c) follows from a) and b).