The representation dimension $\rdim(G)$ of a finite group $G$ is the smallest
positive integer $m$ for which there exists an embedding of $G$ in $\GL_m(\C)$.
In this paper we find the largest value of $\rdim(G)$, as $G$ ranges over all
groups of order $p^n$, for a fixed prime $p$ and a fixed exponent $n \ge 1$.

Let $k$ be a field of characteristic zero, let $G$ be a connected reductive
algebraic group over $k$ and let $\mathfrak{g}$ be its Lie algebra. Let $k(G)$,
respectively, $k(\mathfrak{g})$, be the field of $k$-rational functions on $G$,
respectively, $\mathfrak{g}$. The conjugation action of $G$ on itself induces
the adjoint action of $G$ on $\mathfrak{g}$. We investigate the question
whether or not the field extensions $k(G)/k(G)^G$ and
$k(\mathfrak{g})/k(\mathfrak{g})^G$ are purely transcendental.