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A group is cyclic if it can be generated by a single element, $a$. Every element has the form $a^i$ for some $i\in\mathbb{Z}$. As $a^ia^j=a^{i+j}=a^ja^i$, these groups are abelian.

Cyclic groups can be classified. A finite cyclic group is isomorphic to a group of the form $\mathbb{Z}/n\mathbb{Z}$. That is, it is isomorphic to the integers under addition modulo $n$. Any infinite cyclic group is isomorphic to $(\mathbb{Z}, +)$.

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