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

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

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

Then $\left({U_\C, \times}\right)$ and $\left({\Z'_5, \times_5}\right)$ 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 $\left({U_\C, \times}\right)$ is as follows:

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