# Locally cyclic periodic group

From Groupprops

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: locally cyclic group and periodic group

View other group property conjunctions OR view all group properties

## Definition

A group is termed a **locally cyclic periodic group** if it satisfies the following equivalent conditions:

- 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.

### Equivalence of definitions

`Further information: Equivalence of definitions of locally cyclic periodic group`