Isomorphisms between Additive Group of Integers Modulo 4 and Reduced Residue System Modulo 5 under Multiplication

Theorem

Let $\struct {\Z_4, +_4}$ denote the additive group of integers modulo $4$.

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

There are $2$ (group) isomorphisms from $\struct {\Z_4, +_4}$ onto $\struct {\Z'_5, \times_5}$.

Proof

Let us recall the Cayley table of $\struct {\Z_4, +_4}$:

$\begin{array}{r|rrrr} \struct {\Z_4, +_4} & \eqclass 0 4 & \eqclass 1 4 & \eqclass 2 4 & \eqclass 3 4 \\ \hline \eqclass 0 4 & \eqclass 0 4 & \eqclass 1 4 & \eqclass 2 4 & \eqclass 3 4 \\ \eqclass 1 4 & \eqclass 1 4 & \eqclass 2 4 & \eqclass 3 4 & \eqclass 0 4 \\ \eqclass 2 4 & \eqclass 2 4 & \eqclass 3 4 & \eqclass 0 4 & \eqclass 1 4 \\ \eqclass 3 4 & \eqclass 3 4 & \eqclass 0 4 & \eqclass 1 4 & \eqclass 2 4 \\ \end{array}$
$\begin{array}{r|rrrr} \times_5 & \eqclass 1 5 & \eqclass 2 5 & \eqclass 3 5 & \eqclass 4 5 \\ \hline \eqclass 1 5 & \eqclass 1 5 & \eqclass 2 5 & \eqclass 3 5 & \eqclass 4 5 \\ \eqclass 2 5 & \eqclass 2 5 & \eqclass 4 5 & \eqclass 1 5 & \eqclass 3 5 \\ \eqclass 3 5 & \eqclass 3 5 & \eqclass 1 5 & \eqclass 4 5 & \eqclass 2 5 \\ \eqclass 4 5 & \eqclass 4 5 & \eqclass 3 5 & \eqclass 2 5 & \eqclass 1 5 \\ \end{array}$

By arranging the rows and columns into a different order, its cyclic nature becomes clear:

$\begin{array}{r|rrrr} \times_5 & \eqclass 1 5 & \eqclass 2 5 & \eqclass 4 5 & \eqclass 3 5 \\ \hline \eqclass 1 5 & \eqclass 1 5 & \eqclass 2 5 & \eqclass 4 5 & \eqclass 3 5 \\ \eqclass 2 5 & \eqclass 2 5 & \eqclass 4 5 & \eqclass 3 5 & \eqclass 1 5 \\ \eqclass 4 5 & \eqclass 4 5 & \eqclass 3 5 & \eqclass 1 5 & \eqclass 2 5 \\ \eqclass 3 5 & \eqclass 3 5 & \eqclass 1 5 & \eqclass 2 5 & \eqclass 4 5 \\ \end{array}$

Each of these is the cyclic group of order $4$.

Each has $2$ generators, each of $1$ element.

Hence you can get an isomorphism from $\struct {\Z_4, +_4}$ to $\struct {\Z'_5, \times_5}$ by setting up the mappings:

$\phi: \struct {\Z_4, +_4} \to \struct {\Z'_5, \times_5}: \forall x \in \Z_4: \map \phi x = \begin {cases} \eqclass 4 0 & : x = \eqclass 5 1 \\ \eqclass 4 1 & : x = \eqclass 5 2 \\ \eqclass 4 2 & : x = \eqclass 5 4 \\ \eqclass 4 3 & : x = \eqclass 5 3 \end {cases}$
$\psi: \struct {\Z_4, +_4} \to \struct {\Z'_5, \times_5}: \forall x \in \Z_4: \map \psi x = \begin {cases} \eqclass 4 0 & : x = \eqclass 5 1 \\ \eqclass 4 1 & : x = \eqclass 5 3 \\ \eqclass 4 2 & : x = \eqclass 5 4 \\ \eqclass 4 3 & : x = \eqclass 5 2 \end {cases}$

$\blacksquare$