Are the groups $(\mathbb Z/12\mathbb Z,+)$ and $(\mathbb Z/13\mathbb Z,\times)$ isomorphic?

Question

I know both have the same number of elements so that is not a problem but I am having trouble showing this. What could be a map I can use to do this?

Answer 1

Yes: the group $(\mathbb{Z}_{13}\setminus\{0\},\times)$ is also a cyclic group of order $12$, since it is generated by $2$, which is an element of order $12$. So$$\begin{array}{ccc}(\mathbb Z_{12},+)&\longrightarrow&(\mathbb{Z}_{13}\setminus\{0\},\times)\\n&\mapsto&2^n\end{array}$$is a group isomorphism.

Answer 2

More generally, if $p$ is prime, then the groups $(\mathbb Z/(p-1)\mathbb Z,+)$ and $(\mathbb Z/p\mathbb Z,\times)$ are isomorphic. However, an explicit isomorphism cannot be given in general, because no one knows how to find primitive roots explicitly. See Wikipedia for instance.