# Cyclic iff not a union of proper subgroups

From Groupprops

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

## Proof

### Cyclic implies not a union of proper subgroups

**Given**: A cyclic group with cyclic element .

**To prove**: is not a union of proper subgroups.

**Proof**: Since generates , cannot be contained in any proper subgroup of . Hence, any union of proper subgroups of cannot contain , so 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.