Let be the set of (residue classes of) integers , and let
Show that is a group under multiplication. [You may assume throughout this question that multiplication of matrices is associative.]
Let be the set of 2-dimensional column vectors with entries in . Show that the mapping given by
is a group action.
Let be an element of order . Use the orbit-stabilizer theorem to show that there exist , not both zero, with
Deduce that is conjugate in to the matrix