# Any abelian group is cyclic? [closed]

By theorem, every cyclic group is abelian, but does all abelian group are cyclic? I mean, is there a abelian group that is not cyclical?

## closed as off-topic by Dietrich Burde, Claude Leibovici, Strants, Xam, user190080Feb 28 '18 at 17:57

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• What are some finite groups you know? There are very small counterexamples you should be aware of. – lulu Feb 28 '18 at 13:08
• Try to find an abelian noncyclic group of order 4. What can you say about the order of the non-identity elements? – almagest Feb 28 '18 at 13:09
• Possible duplicate of Abelian group that is non-cyclic – освящение Feb 28 '18 at 13:28

No, consider the abelian group $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ consisting out of the elements $\{(0,0), (1,0), (0,1), (1,1)\}$. The group operation is entrywise addition and reduction modulo $2$. The non-identity elements have order $2$, and hence there is no element of order $4$.
$\mathbb Z_m×\mathbb Z_n$ is abelian but not cyclic when $gcd(m,n)\gt1$...