This is an unformatted preview. Please download the attached document for the original format.April 15 2014 Spring 2014 California State University Fullerton Math 407 Edited by: 1. (a) Let G be a group and let N be a normal subgroup of G. Prove that if M is a subgroup of G/N then there exists H G such that H/N = M . (b) Let G = r s | r4 = e = s2 sr = r3 s and let N = r2 . Prove that G/N has precisely ve subgroups (including G/N and the identity subgroup.) Are any of these subgroups normal? 2. Let G be a group and let Inn(G) denote the set {cx | x G} where cx : G G g xgx1 . The composition law cx cy = cxy makes Inn(G) into a group. Prove that G/Z(G) is isomorphic to Inn(G).