Mathematics 893 Problem Set 3

Suggested reading: Chapter 2 and Section 3.1. Required reading: Note especially the definitions of the lattice of subgroups, Z(G), and A_{4}, which were not done in class, but are needed in some of the questions. The lattice of subgroups is Section 2.5, Z(G) is defined in Section 2.2, and A_{4} is discussed in Section 3.5.

Questions due next Friday (October 7):

1. Given a group G (with operation x · y), define another operation ∘ on the same set G by x∘y = y ·x. Show that (G, ∘) is a group. Give an isomorphism between (G, ·) and (G, ∘).

2. Let ϕ : G → H be a surjective group homomorphism. Prove that ϕ^{−1}(K) ↔ K gives a bijection between subgroups of G that contain ker ϕ and subgroups of H. Show that K ⊴ H if and only if ϕ^{−1}(K) ⊴ G. (Problem 1 on page 65 is part of this question. Do not assume this problem. Give a complete proof.)

3. Page 33 #15. Use this to do problems 5 and 13 on the same page.

4. Page 48 #3, 11, 15.

5. Give the lattice of subgroups of ℤ/30ℤ. (Examples are found in pages 67-70).

6. Page 60 #14 (do any three parts, one of which should be negative)

7. Page 65 #15

8. Page 85 #5, 10, 36

9. Page 111 #7 Notes: We did quotient groups (Section 3.1) before discussing subgroups of ℤ/nℤ. This makes things a bit easier. Later we will think of abelian groups as modules over the ring ℤ of integers. This will give additional insights. Section 2.2 defines a number of subgroups associated to a group action. Some of these will be introduced in class later when needed. It might be helpful if you read about them ahead of time.