Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 19H (Representation Theory)

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 $p$.

The commutator subgroup equals the centre, soFix $\omega =e^{2\pi i/p}$. Every linear character isand these $p^2$ distinct characters exhaust the one-dimensional representations.

Take the abelian subgroupof order $p^2$. For each $c\in {\mathbb{F}}_p^{\times }$, define ${\theta }_c(x,0,z)={\omega }^{cz}$ and form the induced representationIt has degree $[G:A]=p$. Conjugation by representatives $(0,y,0)$ gives $p$ distinct characters of $A$ because $c\ne 0$. The stabilizer of ${\theta }_c$ is therefore exactly $A$, and the Mackey irreducibility criterion shows that ${\rho }_c$ is irreducible. The $p-1$ choices of $c$ have distinct central characters.

With class representatives $(0,0,z)$ and $(x,y,0)$ for $(x,y)\ne (0,0)$, the complete character table is described byfor $a,b\in {\mathbb{F}}_p$, andfor $c\in {\mathbb{F}}_p^{\times }$. There are $p^2+p-1$ rows, matching the number of conjugacy classes, andso no irreducible characters are missing.