# Let G be a group and let N be a normal subgroup of G. Prove that if M is a subgroup of G/N ,

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April 15, 2014

Spring 2014

California State University Fullerton

Math 407

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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 ﬁve

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 → xgx−1 .

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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