Equivalence of definitions of locally cyclic periodic group

Statement
The following are equivalent for a group:


 * 1) It is a fact about::locally cyclic group as well as a fact about::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-fact about::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: