Uniqueness of cyclic groups up to isomorphism. I am reading something about abstract algebra. Can anyone please tell me if the following statements are saying: a cyclic group can only be isomorphic to either $\mathbb{Z}/n\mathbb{Z}$ if it is finite or $\mathbb{Z}$ if it is infinite?
Thanks a lot.
The group $\mathbb{Z}$ is the only infinite cyclic group, up to isomorphism.
The group $\mathbb{Z}/n\mathbb{Z}$ is the only cyclic group of order $n$, up to isomorphism

Yes, that is correct.  If a group $G$ is cyclic, then $G$ is generated by (the positive and negative powers of) a single element $a$, i.e there exists an $a \in G$ such that $G=\{a^n: n \in \mathbb{Z} \}$. If $G$ is finite, then there exists an isomorphism from $G$ to $C_n$, and if $G$ is infinite, then there exists an isomorphism from $G$ to $\mathbb{Z}$. 
