On ℤ, let a R b mean that a - b is divisible by 4.

Step 1. Reflexive. $a - a = 0$ is divisible by $4$.

Step 2. Symmetric. If $4$ divides $a - b$, it divides $b - a = -(a - b)$.

Step 3. Transitive. If $4$ divides $a - b$ and $b - c$, it divides their sum, $a - c$.

Step 4. The group of $5$. All integers that leave a remainder of $1$ on division by $4$: $\{\ldots, -3, 1, 5, 9, 13, \ldots\}$.