Exercise 11, § 2

We denote by L(G) the set of continuous representations of G in $\mathbf{R}$. We call one-parameter subgroup of G the image of a continuous morphism of $\mathbf{R}$ into G.

a) The following conditions are equivalent :

(i) G is the union of one-parameter subgroups ;

(ii) every morphism of $\mathbf{Z}$ into G extends to a continuous morphism of $\mathbf{R}$ into G;

(iii) every continuous morphism of $\widehat{G}$ into $\mathbf{R}/\mathbf{Z}$ is obtained by passing to the quotient from an element of $L(\widehat{G})$;

(iv) every unitary character of $\widehat{G}$ is of the form $e^{i\lambda}$ where $\lambda \in L(\widehat{G})$.

b) If G has a countable basis, the conditions of a) are equivalent to : G is isomorphic to $\mathbf{R}^n\times \mathbf{T}^I$ where I is a countable set.

c) The union of the one-parameter subgroups of G is a dense subgroup in the neutral component of G. (Use exerc. 4.)

d) Let $t\mapsto \chi_t$ be a continuous mapping of $[0,1]$ into $\widehat{G}$ such that $\chi_0$ is the trivial character. There exists a mapping $t\mapsto \lambda_t$ of $[0,1]$ into L(G) such that $\lambda_0= 0$, for every $t$ one has $\chi_t=$ exp(2$i\pi \lambda_t$), and for every $x\in G$, the mapping $t\mapsto \lambda_t(x)$ is continuous.

e) The union of the one-parameter subgroups of G is also the union of the arcs of G which contain $e$, an arc of G being the image of a continuous mapping of $[0,1]$ into G. (Use d).)