# Faithful representations and tensor products

Let $G$ be a finite group (or a compact Lie group). Prove that if $\rho:G\to\mathrm{GL}(V)$ is a faithful finite dimensional complex representation of $G$ then any irreducible representation embeds in some tensor product of $V$.

# Be rational

Most of representation theory results work out for algebraically closed fields. For instance, we can classify irreducible representations for the dihedral group $D_{2n}$ with $2n$ elements over $\mathbb{C}$. Classify all irreducible representations for $D_{2n}$ over $\mathbb{Q}$.

# Fundamental theorem of algebra…?

Let $C_{n}$ be the cyclic group of order $n$ and consider $\mathbb{C}C_n$ the group algebra. Prove that the number of solutions to the equation $x^k = x$ for $x\in\mathbb{C}C_n$ is $k^n$ for $k>1$.

# Some representation theory

Find all representations over $\Bbb{C}$ of the group of matrices

$\begin{bmatrix} a&b\\ 0&a^{-1} \end{bmatrix}$

where $a\neq 0$ and $b$ are in $\Bbb{F}_p$.