Let $p$ be a prime and let $\mathbf{Z}_p$ denote the ring of $p$-adic integers. Suppose that $n$ is an integer with $p>n+1$. Then it's well-known that the general linear group ${\rm GL}_n(\mathbf{Z}_p)$ is $p$-torsion-free. (Indeed, note that there is no element in $\mathbf{Z}_p$ having order $p$ if $p\geq 3$. If ${\rm GL}_n(\mathbf{Z}_p)$ has an element $x$ of order $p$, then the minimal polynomial of $x$ must divide $t^p-1$ and hence it has degree $p-1$. Since the minimal polynomial of $x$ is of degree at most $n$, we see that $p-1\leq n$, which is a contradiction.) It leads to the following question:
What happens for $p=2$? More precisely, denote by $T({\rm GL}_n(\mathbf{Z}_2))$ the set of torsion elements in ${\rm GL}_n(\mathbf{Z}_2)$. When is $T({\rm GL}_n(\mathbf{Z}_2))$ a group? For example, is $T({\rm GL}_n(\mathbf{Z}_2))$ always a group? Or, $T({\rm GL}_n(\mathbf{Z}_2))$ cannot be a group for $n\geq 2$?
When $n=1$, we have $T({\rm GL}_1(\mathbf{Z}_2))=\{\pm 1\}$. Thus, we may assume that $n\geq 2$. Any comments or references would be highly appreciated.