Let $n \in \mathbb{Z}^{+}$, and let $Z_n$ be the cyclic group of order $n$. I understand that $Z_n \cong \mathbb{Z}/n\mathbb{Z}$, but the Dummit and Foote book on Abstract Algebra insists that $Z_n$ is unique. Why is $Z_n$ necessarily unique? Is it because of the isomorphism above? My intuition tells me that we say it is unique because of this isomorphism, all cyclic groups of order $n$ have "the same" structure as integers modulo $n$.
Reference, pg. 56 edition 3.