(a) Let be a prime, and let be the group of matrices of determinant 1 with entries in the field of integers .
(i) Define the action of on by MÃ¶bius transformations. [You need not show that it is a group action.]
State the orbit-stabiliser theorem.
Determine the orbit of and the stabiliser of . Hence compute the order of .
Show that is conjugate to in if , but not if .
(b) Let be the set of all matrices of the form
where . Show that is a subgroup of the group of all invertible real matrices.
Let be the subset of given by matrices with . Show that is a normal subgroup, and that the quotient group is isomorphic to .
Determine the centre of , and identify the quotient group .