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I need help with the following problem:

Suppose $n$ is some positive integer, and $n|p-1$. Classify all irreducible representations of $$\mathbb{Z}/p\mathbb{Z}\rtimes \mathbb{Z}/n \mathbb{Z}.$$

I am not sure where to start,

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Thank you in advance!!

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I assume that you are talking about a faithful action of $\mathbb{Z}/n\mathbb{Z}$ on $\mathbb{Z}/p\mathbb{Z}$. First, there are the obvious lifts of the irreducible representations of $\mathbb{Z}/n\mathbb{Z}$ to $G$. For the rest:

**Exercise 1:** The induction of a non-trivial character of $\mathbb{Z}/p\mathbb{Z}$ to $G$ is irreducible.

**Exercise 2:** When are two such inductions isomorphic?

**Exercise 3:** Now count the sums of squares of the degrees of the characters that you get this way.

I have to say that this is rather tough homework if you were not given any of these hints. But with them, you should be able to do the rest yourself.

*Edit:* More generally, suppose that $G=A\rtimes H$ where $A$ is abelian. Then all irreducible characters of $G$ are obtained as follows. $H$ acts on the irreducible characters of $A$ by $(h\cdot\chi)(a) = \chi(h^{-1}ah)$. Let $\chi$ be a linear character of $A$. Extend it to $S_\chi=A\rtimes \text{Stab}_H(\chi)$, where $\text{Stab}_H(\chi)\leq H$ is the stabiliser of $\chi$ in $H$ under the above action, by setting $\chi(as) = \chi(a)$ for $a\in A,s\in \text{Stab}_H(\chi)$. Let $\rho$ be an irreducible character of $\text{Stab}_H(\chi)$, lift it to an irreducible character of $S_\chi$. Then $\text{Ind}_{G/S_\chi}(\chi\otimes \rho)$ is an irreducible character of $G$ and they all arise in this way. I will leave it to you to determine when two such inductions are isomorphic, so as not to spoil the homework exercise.

Note that your question is a special case of this, since $H=\mathbb{Z}/n\mathbb{Z}$ acts faithfully on $\mathbb{Z}/p\mathbb{Z}$, and therefore also on its irreducible characters. Thus, $\text{Stab}_H(\chi)$ is trivial in your case, whenever $\chi$ is non-trivial.

Have a look at Weintraub – “Representation theory of finite group” in the section Mackey Machine. They give the right tool for exactly studying the irreducible representation of a semidirect product $H \rtimes G$, where $H$ is abelian.

In fact, the Mackey Machine holds in greater generality for an exact sequence of locally compact groups

$ 1 \rightarrow H \rightarrow K \rightarrow G \rightarrow 1,$

with some mild conditions on $H$ (being type 1).

Here is a place to start, although not a complete solution by any means. Let $G=\mathbb{Z}/p\mathbb{Z}\rtimes \mathbb{Z}/n \mathbb{Z}$. Then $\mathbb{Z}/n\subset G$ and $\mathbb{Z}\subset G$.

If we restrict a representation to one of these two subgroups, it will split as a direct sum of irreducibles, and because the subgroups are abelian, this means that we can find a basis of eigenvectors for the action of $\mathbb{Z}/p$ (or a basis of eigenvectors for the action of $\mathbb{Z}/n$).

Suppose that $\omega$ is a $p$th root of unity, and that $v$ is a vector such that $k.v=\omega^k v$ for $k\in \mathbb{Z}/p$. If $j\in \mathbb{Z}/n$, then how does $k$ act on $j.v$? Use the commutation relations that you know you have in $G$ between elements of $\mathbb{Z}/n$ and $\mathbb{Z}/p$

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