Isomorphism between Additive Group Modulo 16 and Multiplicative Group Modulo 17

Theorem
Let $\struct {\Z_{16}, +}$ denote the additive group of integers modulo $16$.

Let $\struct {\Z'_{17}, \times}$ denote the multiplicative group of reduced residues modulo $17$.

Let $\phi: \struct {\Z_{16}, +} \to \struct {\Z'_{17}, \times}$ be the mapping defined as:


 * $\forall \eqclass k {16} \in \struct {\Z_{16}, +}: \map \phi {\eqclass k {16} } = \eqclass {3^k} {17}$

Then $\phi$ is a group isomorphism.