Cyclic Group Z4
Definition
The set of integers $\{ 0, 1, 2, 3 \}$ under modulo-4 addition is called the cyclic group $\mathbb{Z}_{4}$.
Modulo $n$ Addition
Consider the integers $a,b \geq 0$. We can write the addition $a+b$ as
$$ a + b = qn + \beta \quad \text{where} \quad q \in \mathbb{Z} \quad \text{and} \quad \beta \in \{0, 1, \dots, n-1 \} $$And so, we define modulo $n$ addition as
$$ a +_{n} b = \beta $$Multiplication Table
$$ \begin{array}{c|cccc} \mathbb{Z}_{4} & 0 & 1 & 2 & 3 \\ \hline 0 & 0 & 1 & 2 & 3 \\ 1 & 1 & 2 & 3 & 0 \\ 2 & 2 & 3 & 0 & 1 \\ 3 & 3 & 0 & 1 & 2 \end{array} $$Group Properties
Based on the multiplication table, we know that closure, identity and inverse properties are verified.
Associativity
Consider $a,b,c \in \mathbb{Z}_{n}$. We must show that
$$ (a +_{n} b) +_{n} c = a +_{n} (b +_{n} c) $$With $p,q \in \mathbb{Z}$ and $\alpha, \beta \in\mathbb{Z}_{n}$,
$$ \begin{alignat}{2} a + b &= qn + \beta &&\quad\implies a +_{n} b = \beta = a + b - qn \\[6pt] a + b + c &= pn + \alpha &&\quad\implies a +_{n} b +_{n} c = \alpha = a + b + c - pn \end{alignat} $$$$ (a +_{n} b) +_{n} c = \beta +_{n} c = (a+b-qn) +_{n} c $$$$ \begin{align} &(a+b-qn) + c = (p-q)n + \alpha \\ &\implies (a + b - qn) +_{n} c = \alpha \\ &\implies (a+_{n} b) +_{n} c = \alpha \end{align} $$The same process can be repeated to show that $a +_{n} (b +_{n} c) = \alpha$. Therefore, the group multiplication is accociative.
Equivalence Classes
The following equivalence classes were determined by intuition, but they can be verified explicitely with the equivalence definition.
$$ \begin{array}{c|c|c} \mathcal{C}_{1} & \{ 0 \} & \text{trivial identity class} \\[6pt] \hline \mathcal{C}_{3} & \{ 2 \} & \text{center point rotation} \\[6pt] \hline \mathcal{C}_{3} & \{ 1, 3 \} & \text{other rotations} \end{array} $$Subgroups
There is only one subgroup.
$$ H = \{ 0, 2 \} $$The operations can easily be shown to respect closure, thereby making it a subgroup.
Invariant Subgroup
Since $H$ is the union of equivalence classes $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$, $H \triangleleft \mathbb{Z}_{4}$.
Quotient Group
We define the quotient group $H / \mathbb{Z}_{4}$, with the only invariant subgroup $H$, as the sets
$$ H = \{ \{ 0, 2 \}, \{ 1, 3 \} \} $$under group multiplication.
$$ \begin{array}{c|cc} H / \mathbb{Z}_{4} & \{ 0, 2 \} & \{ 1, 3 \} \\[6pt] \hline \{ 0, 2 \} & \{0, 2\} & \{ 1, 3 \} \\[6pt] \{ 1, 3 \} & \{1, 3 \} & \{ 0, 2 \} \end{array} $$