Corollary 6

Let $k\in \mathbf{Z}$. Then the kernel of the homomorphism $x\mapsto x^k$ of G into G and the closure of the image of the morphism $\chi \mapsto \chi^k$ of $\widehat{G}$ into $\widehat{G}$ are orthogonal to one another.

This follows from the preceding Corollary since the morphisms $x\mapsto x^k$ of G into G and $\chi \mapsto \chi^k$ of $\widehat{G}$ into $\widehat{G}$ are dual to one another.

Recall (A, X, p. 17) that a commutative group A is divisible if, for every nonzero $n\in \mathbf{Z}$, the mapping $a\mapsto a^n$ of A into A is surjective.