G is a group given by :
G = { q | ad - bc = 1} where a, b, c, d are integers modulo p, a prime number and
Find the o(G)
let be the group of invertible matrices with entries in define the homomorphism by then and thus
to find look at the elements of as invertible transformations from to where is a vector space of dimension over let be a basis for let
then may be defined to be any non-zero element of so has options. now can be anything in so there are options for
hence and therefore
the above method can be used to, in general, find the order of the group of all matrices with determinant 1 and with entries from some finite field. (just copy the above proof!)