Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 19F (Representation Theory)

The kernel is $\{g:\rho (g)=I\}$ and is normal because $\rho (hg{h}^{-1})=\rho (h)\rho (g)\rho (h{)}^{-1}$. A one-dimensional representation kills every commutator. Conversely, irreducibles killing $G^{\prime }$ factor through the finite abelian group $G/G^{\prime }$ and are one-dimensional.

By Schur lemma, $z\in Z$ acts on irreducible $V$ by a scalar. Induced representation shows that $V$ occurs in ${\mathrm{Ind}}_Z^G\Theta$ iff that scalar character is $\Theta$, and then with multiplicity $\dim V$.

Here $|G|=p^3$, $Z=G^{\prime }=\{(x,y,c):x=y=0\}\simeq C_p$. Degree divisibility and $\sum d_i^2=p^3$ exclude degrees strictly between $1$ and $p$. There are $p^2$ linear characters, corresponding to $G/G^{\prime }$. For trivial $\Theta$, induction is their direct sum. For each of the $p-1$ nontrivial central characters, induction is $p$ copies of one degree-$p$ irreducible $V_{\Theta }$. Thus the complete list is $p^2$ degree-one and $p-1$ degree-$p$ representations; their squared degrees sum to $p^3$.