I will define the following groupoid. $V_4 := \{1, a, b, c\}$, multiplication defined by $1 * a = a * 1 = a, 1 * b = b * 1 = b , 1 * c = c * 1 = c$, $a * a = b * b = c * c = 1$, $a * b = b * a = c$.
(1) Verify this defines a groupoid. In particular, check for three things: $V_4$ has an identity, an inverse and a partially defined multiplication. Draw the multiplication table for $V_4$. Bonus: In fact, demonstrate $V_4$ is a group