# Equivalence of definitions of locally cyclic periodic group

From Groupprops

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

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

## Statement

The following are equivalent for a group:

- It is a Locally cyclic group (?) as well as a Periodic group (?): every element has finite order.
- It is isomorphic to a restricted direct product of groups, where for each prime , there is either one cyclic group of order a power of appearing or one p-Quasicyclic group (?) appearing.

## Proof

**Given** A periodic locally cyclic group .

**To prove**: is isomorphic to a restricted direct product of groups, where for each prime , there is either one cyclic group of order a power of appearing or one p-quasicyclic group appearing.

**Proof**: For each prime , let be the subgroup of comprising elements whose order is a power of . Note that this is a subgroup, because the conditions for products, inverses, and identity element are satisfied.

- is a restricted direct product of the s: Since every element of has finite order, it generates a finite cyclic group which is a direct product of cyclic groups of prime power order. In particular, the element itself is a product of commuting elements of prime power order. Thus, the s generate . Also, each intersects trivially the subgroup generated by all the others, because the order of an element in is a power of and the order of any element generated by the other pieces is relatively prime to .
- Each is locally cyclic and periodic: This follows because both conditions inherit to subgroups.
- If finite, is cyclic: This follows immediately from the previous step, because a finite locally cyclic group is cyclic.
- If infinite, is the -quasicyclic group, i.e., it is isomorphic to the group of all -powered complex roots of unity under multiplication:
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