2. In this uestion we rove that 54 E 33 and construct an ex licit isomo hism. '1 P V P rp (a) For the factor group above to make sense, V must be a normal subgroup of 54. In this case V = {8; (12) (34); (13) (24); (14) (23)} Explain why V is normal. (b) How many other subgroups does 84 have which are isomorphic to V? Why are none of them normal? (c) Explain why 54/V is a group of order six. (d) By part (c), we must have either 3%; E 26 or 54/1/ g 53. Show that (12)V(13)V 7e (13)V(12)V Why does this prove that 54/V E 53? (e) Why is it obvious that the following six left cosets are all di'erent? (Think about 4...) V, (12)V, (13)V, (23)V, (123)V, (132)V (f) Dene an isomorphism y : 54/17 > 53 and prove that it is an isomorphism