Theorem: Isomorphism between Additive Group Modulo 16 and Multiplicative Group Modulo 17

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:
{{begin-eqn}}
{{eqn | l = \map \phi {\eqclass x {16} } \times \map \phi {\eqclass y {16} }
      | r = \map \phi {x + 16 m_1} \times \map \phi {y + 16 m_2}
      | c = {{Defof|Residue Class}}: for some representative $m_1, m_2 \in \Z$
}}
{{eqn | r = 3 \uparrow \paren {x + 16 m_1} \times 3 \uparrow \paren {y + 16 m_2}
      | c = using Knuth uparrow notation $3 \uparrow k := 3^k$
}}
{{eqn | r = 3 \uparrow \paren {x + 16 m_1 + y + 16 m_2}
      | c = Product of Powers
}}
{{eqn | r = 3 \uparrow \paren {\paren {x + y} + 16 \paren {m_1 + m_2} }
      | c = 
}}
{{eqn | r = 3 \uparrow \paren {\eqclass {x + y} {16} }
      | c = {{Defof|Residue Class}} and {{Defof|Modulo Addition}}
}}
{{eqn | r = \map \phi {\eqclass x {16} + \eqclass y {16} }
      | c = Definition of $\phi$
}}
{{end-eqn}}
Thus it is seen that $\phi$ is a group homomorphism.
{{qed|lemma}}
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.
{{qed|lemma}}
Now we have:
{{begin-eqn}}
{{eqn | l = 16
      | o = \equiv
      | r = 0
      | rr= \pmod {16}
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \map \phi {\eqclass {16} {16} }
      | r = \map \phi {\eqclass 0 {16} }
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \map \phi {\eqclass {16} {16} }
      | r = \eqclass 1 {17}
      | c = Group Homomorphism Preserves Identity
}}
{{eqn | n = 1
      | ll= \leadsto
      | l = 3^{16}
      | o = \equiv
      | r = 1
      | rr= \pmod {17}
      | c = Definition of $\phi$
}}
{{end-eqn}}
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:
{{begin-eqn}}
{{eqn | l = \map \phi {\eqclass x {16} }
      | r = \map \phi {\eqclass y {16} }
      | c = 
}}
{{eqn | ll= \leadsto
      | q = \forall m_1, m_2 \in \Z
      | l = \map \phi {x + 16 m_1}
      | r = \map \phi {y + 16 m_2}
      | c = {{Defof|Residue Class}}
}}
{{eqn | ll= \leadsto
      | l = 3 \uparrow \paren {x + 16 m_1}
      | r = 3 \uparrow \paren {y + 16 m_2}
      | c = Definition of $\phi$
}}
{{eqn | ll= \leadsto
      | l = 3^x \paren {3^{16} }^{m_1}
      | r = 3^y \paren {3^{16} }^{m_2}
      | c = Product of Powers, Power of Power
}}
{{eqn | ll= \leadsto
      | l = 3^x \times 1^{m_1}
      | r = 3^y \times 1^{m_2}
      | c = as $3^{16} \equiv 1 \pmod {17}$ from $(1)$
}}
{{eqn | ll= \leadsto
      | l = 3^x
      | r = 3^y
      | c = 
}}
{{eqn | ll= \leadsto
      | l = x
      | r = y
      | c = 
}}
{{end-eqn}}
Thus $\phi$ is an injection.
From Equivalence of Mappings between Finite Sets of Same Cardinality it follows that $\phi$ is a bijection.
{{qed|lemma}}
Thus $\phi$ is a bijective group homomorphism.
Hence the result by definition of group isomorphism.
{{Qed}}