# Locally cyclic iff subquotient of rationals

From Groupprops

This article gives a proof/explanation of the equivalence of multiple definitions for the term locally cyclic group

View a complete list of pages giving proofs of equivalence of definitions

## Contents

## Statement

The following are equivalent for a group:

- It is a locally cyclic group: every finitely generated subgroup of the group is cyclic.
- It is isomorphic to a subquotient (i.e., a quotient group of a subgroup) of the group of rational numbers.

## Facts used

- Locally cyclic implies periodic or aperiodic: In a locally cyclic group, either all the element have finite order, or all non-identity elements have infinite order.
- Equivalence of definitions of locally cyclic aperiodic group: A group is locally cyclic and aperiodic iff it is isomorphic to a subgroup of the group of rational numbers.
- Equivalence of definitions of locally cyclic periodic group: A locally cyclic periodic group is a restricted direct product of groups indexed by distinct primes , where the group indexed by a prime is either cyclic of order a power of or a -quasicyclic group.

## Proof: Subquotient of rationals implies locally cyclic

To prove this, it suffices to note the following things:

- The group of rational numbers is locally cyclic.
- Local cyclicity is subgroup-closed: Any subgroup of a locally cyclic group is locally cyclic.
- Local cyclicity is quotient-closed: Any quotient of a locally cyclic group is locally cyclic.

## Proof: locally cyclic implies subquotient of rationals

**Given**: A locally cyclic group .

**To prove**: is isomorphic to a subquotient of the group of rational numbers.

**Proof**:

- By fact (1), is either periodic or aperiodic.
- If is aperiodic, it is isomorphic to a subgroup of the group of rational numbers by fact (2), and hence to a subquotient of the group of rational numbers. This completes the proof of the aperiodic case.
- If is periodic, then by fact (3), it is a restricted direct product of groups indexed by distinct primes , where the group indexed by a prime is either cyclic of order a power of or a -quasicyclic group. We argue that a restricted direct product of the form proved above is isomorphic to a subgroup of . Indeed, each is isomorphic to a subgroup: when finite cyclic of order , it is the set of elements of order ; when infinite, it is the set of all elements whose order is a power of . The subgroup these generate inside is readily seen to be isomorphic to .