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I.S.I. and C.M.I. Entrance

Order of General and Special Linear Group

Here is the post in which you would learn about the Order of General and Special Linear Group with the help of a problem. Try it and learn the solution.

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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By Ashani Dasgupta

Founder Director at Cheenta
Pursuing Ph.D. in Mathematics from University of Wisconsin Milwaukee
Research Interest - Geometric Topology

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