Higher Algebra I — Fall 2005

Gerald Hoehn

Problem sheet 2 September 1, 2005

Problem 1: Prove the following remark from the lecture about the centralizer and normalizer of a subset S of a group G:

(a) Cent(S) < Nor(S) < G.

(b) H < G implies H ◁ Nor(H).

(c) K < G and H ◁ K implies K < Nor(H), i.e., Nor(H) is the largest subgroup of G in which H is normal.

Problem 2: Let c : G ⟶ Perm(G) be the map defined for a ∈ G by c(a) : G → G, g ↦ aga^{−1}.

(a) Show that c is a homomorphism from G into Aut(G) < Perm(G), the subgroup of automorphisms of G.

(b) Is the image of c normal in Aut(G)?

(c) Determine the kernel of c.

Problem 3: (a) Let G be group and H be a group of finite index. Show that there exists a normal subgroup N of G contained in H and also of finite index. (Hint: If [G : H] = n, find a homomorphism of G into S_{n} (the group of permutations of the set {1, …, n}) whose kernel is contained in H.)

(b) Let G be a group and let H_{1}, H_{2} be subgroups of finite index. Prove that H_{1} ∩ H_{2} has finite index.

Problem 4: Determine a composition series of the group S_{4}, the group of permutations of the set {1, 2, 3, 4}.