Cyclicity is not finite direct product-closed

This article gives the statement, and possibly proof, of a basic fact in group theory.
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Statement

It is possible to have cyclic groups $G_{1}$ and $G_{2}$ such that the external direct product $G_{1}\times G_{2}$ is not a cyclic group.

Proof

An example: the direct product of cyclic group:Z2 and cyclic group:Z2 is the Klein four-group, which is not cyclic.

Stronger statements

In fact, if both $G_{1}$ and $G_{2}$ are nontrivial finite cyclic groups and their orders are not relatively prime to each other, or if one of them is infinite, the direct product will not be cyclic. See Direct product of cyclic groups is cyclic iff their orders are relatively prime.