Order of General and Special Linear Group

Let p be a prime number. Let G be the group of all $latex 2 times 2 $ matrices over $latex Z_p $ with determinant 1 under matrix multiplication. Then the order of G is

1. (p-1)p(p+1);
2. $latex p^2 (p-1)$
3. $latex p^3 $
4. $latex p^2 (p-1) + p $

(Additional Problem - this did not come in IIT JAM 2013): Find the order of general linear group $latex GL_2 (F_p) $.

Solution:

First we find the order of General Linear Group $latex GL_2 (F_p) $ ; it is the group of all $latex 2 times 2 $ matrices with elements from a prime field (matrix multiplication defined accordingly) which are invertible.

$latex F_p $ has p elements (0, 1, 2, ... , p-1) hence to build a $latex 2 times 2 $ matrix there are p choices for each of the 4 spots creating $latex p^4 $ matrices in total. Since invertible matrices have non zero determinant, we deduct the number of matrices with zero determinant from $latex p^4 $ to get the order of $latex GL_2 (F_p) $.

Suppose a matrix with zero determinant is represented by $latex

Other useful links:-

• https://www.cheenta.in/fermat-point/
• https://www.youtube.com/watch?v=zEIAqPQnoVk&pbjreload=10