Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 19H / a / Solution

Write an element as $(x,y,z)$, with multiplicationThe centre consists of $(0,0,z)$. Each central element is its own conjugacy class. If $(x,y)\ne (0,0)$, conjugation leaves $x,y$ fixed and changes $z$ by every element of ${\mathbb{F}}_p$, so its class isThus there are $p$ central singleton classes and $p^2-1$ noncentral classes of size $p$.