Question

Let G be the group of all 2 x 2 matrices ca 2) where a, b, c, d d are integers modulo pp a prime number, such that ad - bc + 0. G forms a group relative to matrix multiplication. What is o(G)? Let H be the subgroup of the G of part (a) defined by H-{C :) e Gad - bc = 1 | ad – be = 1 What is o(H)? ​

Answer 1

Step-by-step explanation:

Hint: You should prove the following:

Multiplication of matrices ⋅:G×G→G:(g,h)↦g⋅h is an operation, i.e., the image lies in G.

Multiplication of matrices is associative and the identity matrix is the unit element. That's clear.

Finally, you need to show that each matrix a∈G has an inverse in G.