# Representations of finite group with normal prime index subgroup

The problem is this:

Let $$G$$ be a finite group, $$Q$$ a normal subgroup of prime index.

(i) Let $$\psi = \chi_V$$ be the character of the $$Q$$ representation $$V$$. Show that either $$\psi$$ can be extended to $$G$$, or the induced representation is irreducible.

(ii) Show that if $$Q$$ is abelian, then every irreducible representation of $$G$$ is either one-dimensional or $$p$$-dimensional

For the first part, I've done one direction: If $$V = \text{Res}(U)$$, then $$\text{Ind}(V)$$ cannot be irreducible in this case - using Frobenius reciprocity and the Frobenius character formula. However, I'm stuck on the reverse direction.

For the second part, I have a strategy I'm a little bit skeptical of. Basically, I'm thinking you take an irrep $$V$$ of $$G$$, and restrict to $$Q$$. If this is an irreducible $$Q$$ rep, then we're done. Else, it has an irreducible $$Q$$ subrep $$U$$. I'm thinking that if I can show that $$q \in Q$$, $$q \not = e$$, then $$gU \neq U$$. In this case, I'm thinking I can then exhibit $${\rm Ind}(U)$$ as a $$G$$ subrep of $$V$$, and then I would be done since it must be the whole of $$V$$. But, I'm pretty skeptical this totally works since I neither used the previous part, not the hypothesis that $$Q$$ is normal of prime index.

If someone could give me some insight into this question, I would appreciate it. I would also appreciate any general advice on dealing with the relationship between representations of some group $$G$$, and the representations of subgroups and quotients of $$G$$.

• Are you familiar with Clifford theory?
– Nate
Mar 25, 2020 at 17:24
• Sadly no, is it relevant?
– msm
Mar 25, 2020 at 19:55
• Clifford theory pertains to induction and restriction to and from normal subgroups, and it's very useful for problems like these. (Try google-ing it)
– Nate
Mar 25, 2020 at 21:45

Write $$G/Q = \langle \sigma Q\rangle$$ for some element $$\sigma\in G$$. Let $$\rho = \mathrm{Ind}_Q^G(\psi)$$. By Frobenius reciprocity, $$(\rho, \rho)_G = (\psi, \mathrm{Res}^G_Q(\mathrm{Ind}_Q^G(\psi))_Q = (\psi, \bigoplus_{i = 0}^{p-1}\psi^{\sigma^i})_Q.$$ Here, the final equality follows from Mackey theory. If $$\psi\simeq \psi^{\sigma^i}$$ for some $$i\ne 0$$, then, since $$\sigma$$ has prime order, $$\psi\simeq \psi^{\sigma^i}$$ for all $$i$$. Hence, $$(\rho, \rho)_G$$ is either $$1$$ or $$p$$. Now use Schur's Lemma to match each of these cases to one of the two options.
Your strategy almost works. The additional fact you need is that $$U$$, as an irreducible finite dimensional representation of an abelian group, is one-dimensional. Now the result follows from part (i), Frobenius reciprocity and Schur's Lemma.