Locally cyclic implies periodic or torsion-free

Statement
A fact about::locally cyclic group (i.e., a group in which every finitely generated subgroup is cyclic) is either a fact about::periodic group (i.e., every element has finite order) or a fact about::torsion-free group (i.e., every non-identity element has infinite order).

Related facts

 * Locally cyclic iff subquotient of rationals
 * Equivalence of definitions of locally cyclic periodic group
 * Equivalence of definitions of locally cyclic torsion-free group

Proof
Given: A locally cyclic group $$G$$.

To prove: For any non-identity elements $$g,h \in G$$, either both $$g$$ and $$h$$ have finite order or both have infinite order.

Proof: Consider the subgroup $$\langle g,h \rangle$$. Since $$G$$ is locally cyclic, there exists $$a \in G$$ such that $$\langle g,h \rangle = \langle a \rangle$$.

We consider two cases:


 * 1) $$a$$ has infinite order: Both $$g$$ and $$h$$ must have infinite order, since they are both finite powers of $$a$$. We can think of them as nonzero elements of the group of integers, with $$a$$ identified with $$1$$.
 * 2) $$a$$ has finite order: Both $$g$$ and $$h$$ must have finite order, and in fact, their orders must divide the order of $$a$$. We can think of them as elements of the finite cyclic subgroup generated by $$a$$.