Locally cyclic iff subquotient of rationals

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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

Statement

The following are equivalent for a group:

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

Facts used

  1. 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.
  2. 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.
  3. Equivalence of definitions of locally cyclic periodic group: A locally cyclic periodic group is a restricted direct product of groups indexed by distinct primes p, where the group indexed by a prime p is either cyclic of order a power of p or a p-quasicyclic group.


Proof: Subquotient of rationals implies locally cyclic

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

  1. The group of rational numbers is locally cyclic.
  2. Local cyclicity is subgroup-closed: Any subgroup of a locally cyclic group is locally cyclic.
  3. 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 G.

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

Proof:

  1. By fact (1), G is either periodic or aperiodic.
  2. If G 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.
  3. If G is periodic, then by fact (3), it is a restricted direct product of groups indexed by distinct primes p, where the group indexed by a prime p is either cyclic of order a power of p or a p-quasicyclic group. We argue that a restricted direct product of the form proved above is isomorphic to a subgroup of \mathbb{Q}/\mathbb{Z}. Indeed, each G_p is isomorphic to a subgroup: when finite cyclic of order p^k, it is the set of elements of order p^k; when infinite, it is the set of all elements whose order is a power of p. The subgroup these generate inside \mathbb{Q}/\mathbb{Z} is readily seen to be isomorphic to G.