Is the following true for some prime $p$?


There exists some prime $\ell$ and some $n$ such that $PSL\_n(\mathbb{F}\_{\ell})$ contains nontrivial $p$-torsion, and moreover if $x \in PSL\_n(\mathbb{F}\_{\ell})$ has order $p$ and $0 < k < p$, then $x$ is conjugate to $x^k$.



Conjugacy of powers of elements in $PSL_n(\mathbb{F}_{\ell})$
Yes. Take $\ell$ to be a primitive root mod $p$ (which exists by Dirichlet's theorem), $n=p-1$, then because $p$ divides $\ell^{p-1}-1$ there is a $p$-torsion element in $PSL\_n(\mathbb F\_\ell$. However, because $p$ is prime to $\ell$, all such elements are semisimple, hence conjugate to their $\ell^k$th powers for any $k$. Because $\ell$ is a primitive root, these powers fill out all the conjugacy classes mod $p$.


