Cyclic iff not a union of proper subgroups

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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

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.