How is it possible that a cyclic group like $\mathbb Z_{2p}$ is isomorphic to a non-cyclic group of order $2p$?

Question

We know that a non-cyclic group can not be isomorphic to a cyclic group. But in Gallian there is a theorem which states that all non-cyclic groups of order $2p$ (where $p$ is prime) are isomorphic to $\mathbb Z_{2p}$ or $D_p$.

Since $\mathbb Z_n$ is a cyclic group, how can it be isomorphic to a non-cyclic group?

Answer 1

It is not possible. Let $f$ be a homomorphism from $\mathbb{Z}_n$ to a group $G$. If it is an isomorphism then the image of the generator of $\mathbb{Z}_n$, for example, $f(1)$ is a generator of $G$ as well.