Abelian not implies cyclic

This article gives the statement, and possibly proof, of a basic fact in group theory.
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Statement

An abelian group is not necessarily a cyclic group.

Proof

The smallest non-cyclic abelian group is the Klein four-group, a group of order 4.

There are infinitely many examples. The elementary abelian groups of a prime power $p^a$ with $a\ge 2$ is a family of abelian non-cyclic groups.

See also

• Cyclic implies abelian