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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

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