Math 6310 • Algebra

Let \(G\) be a finite subgroup of the group \(\operatorname{GL}_n(\mathbb{C})\) of invertible \(n\) by
\(n\) matrices with entries in \(\mathbb{C}\).
\begin{enumerate}
\item[(a)] Show that every element of \(G\) is diagonalizable over \(\mathbb{C}\).
\item[(b)] Give an example of a finite subgroup of \(\operatorname{GL}_n(\mathbb{Q})\) which
contains elements which are not diagonalizable over \(\mathbb{Q}\).
\end{enumerate}