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Contents 1 Statement 2 Related facts 3 Facts used 4 Proof 4.1 Cyclic implies not a union of proper subgroups 4.2 Not cyclic implies 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
View other elementary non-basic facts
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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.