Cyclic iff not a union of proper subgroups

This article describes an easy-to-prove fact about basic notions in group theory, that is not very well-known or important in itself
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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. Every group is a union of cyclic subgroups

Proof

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.