This is an unformatted preview. Please download the attached document for the or

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).

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