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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- 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
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.