I think I have understood that the group $SU(N)$ can be represented by $N\times N$ matrices with unit determinant, and why it has $N^2 -1$ generators. However I dont see why the rank of $SU(N)$ is $N-1$, how can this generally be proved? Also, why is it that the rank of $SU(N)$ gives the number of operators in the algebra that can simultaneously be diagonalized and what does it mean from a physics point of view?