Cardinality of $GL_n(Z_{p^{m}})$

Question

Let $p$ be a prime number and $Z_{p^{m}}$ be the ring of integers modulo $p^m.$ Now one may consider a map $f:Z_{p^{m}} \to Z_p$ which clearly induces a map $F:GL_n(Z_{p^{m}}) \to GL_n(Z_p)$ because of the fact that for any integer $a,$ $gcd(a,p^m)=1$ iff $gcd(a,p)=1.$ Clearly $F$ is given by $F((a_{ij})_{n})=(f(a_{ij}))_n.$ It is also clear that $F$ is surjective. But I want to know the cardinality of $GL_n(Z_{p^{m}}),$ for this I have to know the cardinality of $Ker(F).$ I need some help to calculate the cardinality of $Ker(F).$ Note that class of the elements $\{1,p+1,2p+1,\ldots,p^m-p+1\}$ in $Z_p$ is $1$ and class of the elements $\{0,p,2p,\ldots,p^m-p\}$ in $Z_p$ is $0.$

Thanks

Answer 1

The kernel consists of those elements $M\in\textrm{GL}_n(Z_{p^m})$ that are congruent to the identity $I$ modulo $p$. There are $p^{m-1}$ choices for each entry of $M$, the off-diagonal entries must lie in $pZ_{p^m}$ and the diagonal entries msu lie in $1+pZ_{p^m}$.

Therefore the kernel has order $p^{(m-1)n^2}$.