Exercise 2, § 4

Let G be a totally ordered group. (a) For every element $a > 0$ of G, let $H(a)$ be the smallest isolated subgroup containing $a$, $H^-(a)$ the greatest isolated subgroup not containing $a$; show that $H(a)/H^-(a)$ is isomorphic to a subgroup of $\mathbf{R}$.

(b) Conversely, if H is an isolated subgroup of G such that there exists a greatest element (with respect to inclusion) $H'$ in the set of isolated subgroups of $G$ contained in H and $\neq H$, then $H = H(a)$ and $H' = H^-(a)$ for all $a \in H_+ \cap CH'_+$. Such an isolated subgroup H is called principal and $H'$ is called its predecessor.

(c) If $H_1, H_2$ are two distinct isolated subgroups of $G$ such that $H_1 \subset H_2$, show that there exist two isolated subgroups $H, H'$ of $G$ such that $H_1 \subset H' \subset H \subset H_2$ and $H/H'$ is isomorphic to a non-zero subgroup of $\mathbf{R}$.

(d) Let $\Sigma$ be the set (totally ordered with respect to inclusion) of isolated subgroups of $G$ and let $\Theta \subset \Sigma$ be the set of principal isolated subgroups. Show that $\Sigma$ is isomorphic to the completion (Set Theory, Chapter III, § 1, Exercise 15) of $O$.