Proposition 1

The following conditions are equivalent:

(i) G is generated by a compact subset;

(ii) there exist positive integers $p$ and $q$ and a compact group K such that G is isomorphic to $\mathbf{R}^p\times \mathbf{Z}^q\times K$;

(iii) there exists an integer $n\geqslant 0$ such that $\widehat{G}$ is locally isomorphic to $\mathbf{R}^n$;

(iv) there exist positive integers $p$ and $q$ and a discrete group D such that $\widehat{G}$ is isomorphic to $\mathbf{R}^p\times \mathbf{T}^q\times D$.

(i) $=\Rightarrow$ (iii) : if G has property (i), there exists an integer $n\geqslant 0$ and a subgroup D of G isomorphic to $\mathbf{Z}^n$ such that $G/D$ is compact (lemma 2). Then $D^{\bot}$, which identifies with the dual of $G/D$, is discrete (Theorem 4 of II, p. 226 and prop. 18 of II, p. 233). Hence $\widehat{G}$ is locally isomorphic to $\widehat{G}/D^{\bot}$, that is to say to $\widehat{D}$, which is isomorphic to $\mathbf{T}^n$ (Theorem 4 of II, p. 226 and prop. 18 of II, p. 233). Now $\mathbf{T}^n$ is locally isomorphic to $\mathbf{R}^n$.

(iii) $=\Rightarrow$ (iv) : if $\widehat{G}$ is locally isomorphic to $\mathbf{R}^n$, there exists an integer $p$ such that $0\leqslant p\leqslant n$ such that the identity component $\widehat{G}_0$ of $\widehat{G}$ is an open subgroup isomorphic to $\mathbf{R}^p\times \mathbf{T}^{n-p}$ (TG, VII, p. 13, Theorem 1). In particular, $\widehat{G}_0$ is a divisible group. We then apply lemma 3 to the identity mapping of the subgroup $\widehat{G}_0$ of the group $\widehat{G}$ into the divisible group $\widehat{G}_0$. There therefore exists a morphism $\pi$ of $\widehat{G}$ into $\widehat{G}_0$ which is the identity mapping on $\widehat{G}_0$. Consequently, we have $\pi \circ \pi =\pi$, and $\pi$ is a projector. It is continuous, since its restriction to the open subgroup $\widehat{G}_0$ is so. It follows that $\widehat{G}$ is the direct product of $\widehat{G}_0$ and the subgroup $\overset{-1}{\pi}(e)$, which is discrete since it is isomorphic to $\widehat{G}/\widehat{G}_0$ (TG, III, p. 47, cor.)

(iv) $=\Rightarrow$ (ii) : follows from prop. 2 of II, p. 206, from prop. 18 of II, p. 233 and from corollary 3 of II, p. 236.

(ii) $=\Rightarrow$ (i) : for every compact group K, the group $\mathbf{R}^p\times \mathbf{Z}^q\times K$ is generated by the compact set $[0,1]^p\times \{0,1\}^q\times K$.