Equivalence of definitions of locally cyclic torsion-free group

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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 (?).

Facts used

  1. Local cyclicity is subgroup-closed
  2. Torsion-freeness is subgroup-closed

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.