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

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.