Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 19I (Representation Theory)

For an $H$-module $W$,The action is left multiplication on the first factor:It is well defined because the tensor relation $gh\otimes w=g\otimes hw$ is preserved by left multiplication.

If the $H$-classes in $H\cap \mathcal{C}(g)$ have representatives $x_i$, the induced-character formula isIt follows by grouping the standard sum over $x\in G$ satisfying $x^{-1}gx\in H$.

The subgroup $⟨a⟩$ is the unique Sylow-seven subgroup and is normal. No element outside it centralizes $a$, so $|C_G(a)|=7$. Similarly $C_G(b)=⟨b⟩$ has order three. Thus nonidentity powers of $a$ form two classes of size three, while the elements of orders three form two classes of size seven. Together with the identity, there are five conjugacy classes.

The commutator subgroup is $⟨a⟩$, soHence the three linear characters are

Let $\zeta =e^{2\pi i/7}$ and let ${\lambda }_k(a)={\zeta }^k$. Conjugation by $b$ partitions the six nontrivial characters of $H=⟨a⟩$ into the two orbitsInducing one representative from each orbit gives two irreducible degree-three characters $\psi ,\overline{\psi }$. On the five classes represented by $1,a,a^3,b,b^2$ their values arewith the two cyclotomic sums interchanged for $\overline{\psi }$. Their squared degrees, together with the three linear characters, sum toso these are all irreducible characters.