Subgroups of $\operatorname{GL}(n^2,\mathbf{C})$ containing $\operatorname{PSU}(n)$

Let ${\operatorname{PSU}}(n)$ be the image of the unitary group ${\operatorname{U}}(n)$ under the representation $x\to axa^{-1}$ on the space $M_n({\mathbf C} )$ of $n$ by $n$ complex matrices. We classify all connected Lie subgroups of ${\operatorname{GL}}(n^2,{\mathbf C} )$ containing ${\operatorname{PSU}}(n)$. We use this result to obtain a description of all abstract overgroups of ${\operatorname{PSU}}(n)$ in ${\operatorname{GL}}(n^2,{\mathbf C} )$.

We apply this classification to solve the problem of describing all invertible linear transformations of $M_n({\mathbf C} )$ which preserve the set of normal matrices. Our results can be applied to solve many other problems of similar nature.