Question

Let p be an prime. Use the action of G = GL₂(Z/pZ) on (Z/pZ)² and the orbit-stabilizer theorem to compute the order of G

(Given an element x ∈ X, the orbit O(x) of x is the subsetO(x)={g·x: g∈G}⊂X. Here, we write g · x for ρ(g)(x). The stabilizer of x ∈ X is the subset Gx ={g∈G: g·x=x}⊂G.
This is a subgroup of G, and the orbit-stabilizer theorem says (in a particular form) that if G is a finite group, then we have |G| = |O(x)| · |Gx|.)

(GL₂(Z/pZ) is the group of invertible 2 × 2-matrices with coefficients in Z/pZ, and it acts on (Z/pZ)² viewed as the space of 2 × 1- column vectors via usual matrix multiplication.

Answer 1