# Order of General and Special Linear Group

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

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

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

Solution:

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

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

Suppose a matrix with zero determinant is represented by \$latex