(i) State and prove the Orbit-Stabilizer Theorem.
Show that if is a finite group of order , then is isomorphic to a subgroup of the symmetric group .
(ii) Let be a group acting on a set with a single orbit, and let be the stabilizer of some element of . Show that the homomorphism given by the action is injective if and only if the intersection of all the conjugates of equals .
(iii) Let denote the quaternion group of order 8 . Show that for every is not isomorphic to a subgroup of .