TITLE: The number of $n\times n$ matrix over integer modulo $p$ field with determinant equal $1$
QUESTION [5 upvotes]: How to count the number of $n\times n$ matrix over integer modulo $p$ field with determinant equal $1$?
I know that the number of invertible matrices is GL$(n,p)$. 
Have any ideas?

REPLY [5 votes]: In terms of group theory (perhaps easier to remember), one can define a surjective homomorphism $$\phi: GL(n,p) \rightarrow \mathbb F_p^{*}$$ by $$\phi(A)=det(A).$$ Obviously $ker(\phi)=SL(n,p)$ and since $|\mathbb F_p^{*}|=p-1$, index$[GL(n,p):SL(n,p)]=p-1$.