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
The following are equivalent for a group:
- 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).
- It is isomorphic to a subgroup of the Group of rational numbers (?).
(1) implies (2)
Given: A locally cyclic torsion-free group .
To prove: is isomorphic to a subgroup of the rationals.
Proof: Let be any non-identity element of . Consider the homomorphism defined as follows: . For any , find any nonzero integer such that (in additive notation) for some integer . Then, set .
- Such and exist: [SHOW MORE]
- is well-defined, i.e., for different choices of that work, is the same: [SHOW MORE]
- is a homomorphism: [SHOW MORE]
- 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 to the rationals, completing the proof.