Exercise 3, § 4

(a) Let $\Theta$ be a totally ordered set and for each $s \in \Theta$ let $A_s$ be a subgroup of $\mathbf{R}$ not reduced to 0. Let

be the subgroup of the product $\prod_{s \in \Theta} A_s$, consisting of the functions $f$ whose support is a well-ordered subset of $@$. We define on $G$ an ordering compatible with its group structure taking the set $G_+$ of elements $\geqslant 0$ to be the set consisting of the function 0 and the functions $f \neq 0$ such that $f(\theta) > 0$ for the least element $\theta$ of the support off. Show that $G$ is a totally ordered group and that $\Theta$ is canonically isomorphic to the set of principal isolated subgroups of $G$ ordered by the relation $\supset$; moreover, if $H(a)$ corresponds to $s \in \Theta$ under this isomorphism, $H(a)/H^-(a)$ is isomorphic to $A_s$.

(b) Consider the subgroup $G'$ of the totally ordered group $\mathbf{Q} \times \mathbf{Q}$ (with the lexicographical ordering) generated by the elements $(p_n^{-1}, n p_n^{-1})$, where $(p_n)$ is the strictly increasing sequence of prime numbers. Show that the only isolated subgroup $H'$ of $G'$ distinct from 0 and $G'$ is the group $\{0\} \times \mathbf{Z}$; but $G'$ is not isomorphic to a subgroup of the product $H' \times (G'/H')$ ordered by the lexicographical ordering.