Exercise 10, § 2

A topological group is said to be monothetic if there exists an element of the group whose powers are dense in the group, and solenoidal if there exists a continuous morphism from $\mathbf{R}$ into the group whose image is dense.

a) With the notations of Exercise 9, the group $\Delta_{\boldsymbol{a}}$ is monothetic.

b) Every compact totally disconnected monothetic group is topologically isomorphic to a group of the form $\Delta_{\boldsymbol{a}}$.

c) Suppose G is compact. In order that G be monothetic, it is necessary and sufficient that $\widehat{G}$ be isomorphic to a subgroup of $\mathbf{T}$ endowed with the discrete topology. d) Suppose G is compact. The following conditions are equivalent:

(i) G is solenoidal ;

(ii) $\widehat{G}$ is isomorphic to a subgroup of $\mathbf{R}$ endowed with the discrete topology ;

(iii) $\widehat{G}$ is without torsion, and Card($\widehat{G}$)$\leqslant \mathfrak{c}$, the power of the continuum ;

(iv) G is a quotient of $(\Sigma_{\boldsymbol{a}})^{\mathfrak{c}}$ with $\boldsymbol{a}= (n+ 2)_{n\geqslant 0}$.

e) If G is compact solenoidal, then G is monothetic.