I have to calculate the order of the group $G=\operatorname{GL}(n,\mathbb{Z}_m)$ with $m=p^k$ for $p$ prime and $k\in \mathbb{N}$.
So I was thinking on the homomorphism $\det: G \rightarrow \mathbb{Z}_m$, and the kernel of $\det$ I think should be $\frac{n(n+1)}{2}$, and then the order of the group $G$ will be $\frac{n(n+1)}{2} ((p^k-1)-(k-1))$. Am I on the right direction or not?