# Equivalence of definitions of locally cyclic torsion-free group

This article gives a proof/explanation of the equivalence of multiple definitions for the term locally cyclic torsion-free group
View a complete list of pages giving proofs of equivalence of definitions

## Statement

The following are equivalent for a group:

1. It is both a Locally cyclic group (?) (i.e., every finitely generated subgroup is cyclic) and an Torsion-free group (?) (i.e., every non-identity element has infinite order).
2. It is isomorphic to a subgroup of the Group of rational numbers (?).

## Proof

### (1) implies (2)

Given: A locally cyclic torsion-free group $G$.

To prove: $G$ is isomorphic to a subgroup of the rationals.

Proof: Let $g$ be any non-identity element of $G$. Consider the homomorphism $\varphi:G \to \mathbb{Q}$ defined as follows: $\varphi(g) = 1$. For any $h \in G$, find any nonzero integer $m$ such that (in additive notation) $mh = ng$ for some integer $n$. Then, set $\varphi(h) = n/m$.

1. Such $m$ and $n$ exist: [SHOW MORE]
2. $\varphi$ is well-defined, i.e., for different choices of $(m,n)$ that work, $n/m$ is the same: [SHOW MORE]
3. $\varphi$ is a homomorphism: [SHOW MORE]
4. $\varphi$ is injective: [SHOW MORE]

### (2) implies (1)

This follows from the fact that the group of rationals is both locally cyclic and torsion-free, and facts (1) and (2). We thus have an injective homomorphism from $G$ to the rationals, completing the proof.