# Equivalence of definitions of locally cyclic periodic group

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:

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

## Proof

Given A periodic locally cyclic group $G$.

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

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

1. $G$ is a restricted direct product of the $G_p$s: Since every element of $G$ 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 $G_p$s generate $G$. Also, each $G_p$ intersects trivially the subgroup generated by all the others, because the order of an element in $G_p$ is a power of $p$ and the order of any element generated by the other pieces is relatively prime to $p$.
2. Each $G_p$ is locally cyclic and periodic: This follows because both conditions inherit to subgroups.
3. If finite, $G_p$ is cyclic: This follows immediately from the previous step, because a finite locally cyclic group is cyclic.
4. If infinite, $G_p$ is the $p$-quasicyclic group, i.e., it is isomorphic to the group of all $p^{th}$-powered complex roots of unity under multiplication: PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]