I need a general formula for the order of the group GL( n, Z/p ) of invertible nxn matrices with entries in Z/p, under (matrix) multiplication. I got that, for n=2 the order is (p^2 -1)(p^2 - p), for n=3 the order is (p^3 - 1)(p^3 - p)(p^3 - p^2) ... for n the order is (p^n - 1)(p^n - p)...(p^n - p^(n-1)) (I think?) which is p^(n(n-1)/2) * PRODUCT_(i from 1 to n) [p^i - 1] Is that correct? I'm not sure about it. Also, the number of Sylow p-subgroups in each case is n=2 => (p+1) Sylow p-subgroups where the order is p^(2(2-1)/2) = p^1 n=3 => there will be either 1 or (p+1) or (p^2 + p + 1) Sylow p-subgroups where the order is p^3 general case for n => no idea how many... Is it even possible to describe all those Sylow p-subgroups (which will probably be very complicated)? The only thing I know is that they will be of order p^(n(n-1)/2), given my guess was correct. Can anyone give some help? Thanks.