Let $p$ be a prime and let $\mathbf{Z}_p$ denote the ring of $p$-adic integers. Then there are infinitely many $p$-elements (i.e. those elements of order of a power of $p$) in ${\rm GL}_n(\mathbf{Z}_p)$.
Question: let $\ell $ be an odd prime which is different from $p$. Are there only finitely many $\ell$ elements in ${\rm GL}_n(\mathbf{Z}_p)$?
It is true for $n=1$.