I think I have understood that the group SU(N) can be represented by N×N matrices with unit determinant, and why it has N2−1 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?