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.

Proof
Let $\eqclass x {16}, \eqclass y {16} \in \struct {\Z_{16}, +}$.

Then:

Thus it is seen that $\phi$ is a group homomorphism.

It remains to be seen that $\phi$ is a bijection.

Because $17$ is prime: $\forall x \in \Z, 1 \le x < 17: x \perp 17$ where $\perp$ denotes coprimality.

Thus by definition of multiplicative group of reduced residues modulo $17$:
 * $\order {\struct {\Z'_{17}, \times} } = 16$

where $\order {\, \cdot \,}$ denotes the order of a group.

Similarly, by definition of additive group of integers modulo $16$:
 * $\order {\struct {\Z_{16}, +} } = 16$

So:
 * $\order {\struct {\Z'_{17}, \times} } = \order {\struct {\Z_{16}, +} }$

which is a necessary condition for group isomorphism.

Now we have:

Now let $\eqclass x {16}, \eqclass y {16} \in \Z_{16}$ such that $\map \phi {\eqclass x {16} } = \map \phi {\eqclass y {16} }$.

We have:

Thus $\phi$ is an injection.

From Equivalence of Mappings between Finite Sets of Same Cardinality it follows that $\phi$ is a bijection.

Thus $\phi$ is a bijective group homomorphism.

Hence the result by definition of group isomorphism.