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:

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 $p$ , 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.

$G$ 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 $p$ .
Each $G_{p}$ is locally cyclic and periodic: This follows because both conditions inherit to subgroups.
If finite, $G_{p}$ is cyclic: This follows immediately from the previous step, because a finite locally cyclic group is cyclic.
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]
One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.