# The orders of GL_2(F_3) and SL_2(F_3)

1. May 30, 2009

### moomookow

1. The problem statement, all variables and given/known data
How to find the orders of $$GL_{2}(F_{3}) \, and \, SL_{2}(F_{3})$$, where F_3 is the field Z/3Z.

2. Relevant equations

3. The attempt at a solution
I cannot see a proper way to do this, so I just eliminate all those matrices which are not inversible from all 2x2 matrices over F_3. And I got 52 inversible matrices. and the orders of SL(F) should be 26. Since det:GL(F)->{1,2} and considering some properties of ker(det) and the coset.
I'm not quite confident in my solution, since calculation could be messed up. Is there some better way to solve it?
Thanks

Last edited: May 30, 2009