(a) Let be a finite group. Show that there exists an injective homomorphism to a symmetric group, for some set .
(b) Let be the full group of symmetries of the cube, and the set of edges of the cube.
Show that acts transitively on , and determine the stabiliser of an element of . Hence determine the order of .
Show that the action of on defines an injective homomorphism to the group of permutations of , and determine the number of cosets of in .
Is a normal subgroup of Prove your answer.