Cyclic iff not a union of proper subgroups

Statement
A group is cyclic if and only if it cannot be expressed as a union of proper subgroups. (Note that the trivial group is considered cyclic here).

Related facts

 * Every group is a union of cyclic subgroups
 * Every group is a union of maximal among Abelian subgroups
 * Cyclic of prime power order iff not generated by proper subgroups
 * No proper nontrivial subgroup implies cyclic of prime order

Facts used

 * 1) uses::Every group is a union of cyclic subgroups

Cyclic implies not a union of proper subgroups
Given: A cyclic group $$G$$ with cyclic element $$g$$.

To prove: $$G$$ is not a union of proper subgroups.

Proof: Since $$g$$ generates $$G$$, $$g$$ cannot be contained in any proper subgroup of $$G$$. Hence, any union of proper subgroups of $$G$$ cannot contain $$G$$, so $$G$$ is not a union of proper subgroups.

Not cyclic implies a union of proper subgroups
By fact (1), any group can be expressed as a union of cyclic subgroups. If the group is not itself cyclic, then all these cyclic subgroups are proper, so any non-cyclic group is a union of proper subgroups.