Exercise 20, § 1

Let $G$ be an additive group and let $P$ be a subset of $G$ satisfying the following conditions: 1. $P + P = P$; 2. $P \cap (-P) = (0)$; 3. for every integer $n > 0$, $nx \in P$ implies $x \in P$ (conditions (C)).

EXERCISES

a) If a is an element of G such that a $\notin P$, show that there exists a subset P' of G, satisfying conditions (C'), such that $P \subset P'$ and $-a \in P'$ (take P' to be the set of $x \in G$ such that there exist integers $m > 0$ and $n \geq 0$ and an element $y \in P$ with $mx = -na + y$.

b) Deduce from a) that P is the intersection of those subsets T of G such that $T + T = T, T \cap (-T) = \{0\}, T \cup (-T) = G$ (that is, making G a totally ordered group) and $P \subset T$ (use Zorn's Lemma).

c) In particular if G is an additive group in which every nonzero element has infinite order, then the intersection of all the subsets T of G such that $T + T = T, T \cap (-T) = (0)$ and $T \cup (-T) = G$ is $\{0\}$.