Let be a ﬁeld. Prove that be the group of 2 x 2 matrices with coefficient in and determinant equal to 1.
Question : If F has q elements, how many elements are in the group ?
I know that .
is a subgroup of the group of units of .
From this I've tried to solve the problem but I'm not able to find a way to count the elements in any of the groups.
because any element except 0 has a multiplicative inverse in .
In other word, there are equivalent classes (or orbits) of .
We thus need to find the number of elements in each orbit.
In an equivalent class we have
There are combinations of .
combinations of because we can swap .
There are therefore elements in its equivalent classes.
Am I right?