Isomorphism between Gaussian Integer Units and Reduced Residue System Modulo 5 under Multiplication

Theorem
Let $\struct {U_\C, \times}$ be the group of Gaussian integer units under complex multiplication.

Let $\struct {\Z'_5, \times_5}$ be the multiplicative group of reduced residues modulo $5$.

Then $\struct {U_\C, \times}$ and $\struct {\Z'_5, \times_5}$ are isomorphic algebraic structures.

Proof
Establish the mapping $f: U_C \to \Z'_5$ as follows:

From Isomorphism by Cayley Table, the two Cayley tables can be compared by eye to ascertain that $f$ is an isomorphism:

Cayley Table of Gaussian Integer Units
The Cayley table for $\struct {U_\C, \times}$ is as follows:

Multiplicative Group of Reduced Residues Modulo $5$
The Cayley table for $\struct {\Z'_5, \times_5}$ is as follows: