Corollary 2

The following conditions are equivalent:

(i) The groups G and $\widehat{G}$ are generated by compact subsets ;

(ii) There exist positive integers $n$ and $m$ such that G is locally isomorphic to $\mathbf{R}^m$ and $\widehat{G}$ to $\mathbf{R}^n$;

(iii) There exist positive integers $p,q$ and $r$ and a finite group A such that G is isomorphic to $\mathbf{R}^p\times \mathbf{T}^q\times \mathbf{Z}^r\times A$ ;

(iv) There exist positive integers $p,q$ and $r$ and a finite group A such that $\widehat{G}$ is isomorphic to a product $\mathbf{R}^p\times \mathbf{Z}^q\times \mathbf{T}^r\times A$.

One has (i) $\Leftrightarrow$ (ii) according to prop. 1, and (iii) $\Leftrightarrow$ (iv) by duality, hence (iii) $\Rightarrow$ (i). Finally, if (i) is true, then $\widehat{G}$ is isomorphic to $\mathbf{R}^p\times \mathbf{T}^r\times D$ where D is discrete (prop. 1). The group D is generated by a compact subset, hence finite, of D. Consequently, there exists $q\geqslant 0$ such that D is isomorphic to $\mathbf{Z}^q\times A$, where A is a finite group (A, VII, p. 22, th. 3).