Mathematics 893 Problem Set 2

Suggested reading: Sections 1.2, 1.3, 1.6. We have now covered most of the material in these sections. Required additional reading: Read Section 1.5 on the group Q_{8} (this is used in one of the exercises). Optional reading: (especially if you already have had a good course on group theory). Category Theory, pages 911-912. Note especially the definition of isomorphism in the middle of page 912.

Questions due next Friday (September 30):

1. Page 28, #9,10,11

2. Let G and H be groups and ϕ : G → H be an isomorphism of groups (as defined in the text on page 37). By part (2) of the definition, ϕ is a bijection of sets. Prove that the inverse set map ϕ^{−1} is a group homomorphism.

3. Page 40, #6, 7, 8, 9.

4. Prove that the groups ℚ (under +) and ℚ^{+} (the rational numbers greater than 0 under multiplication) are not isomorphic.

5. Page 33, #2,3

6. The elements of D_{12} can be thought of as permutations of the vertices {1, 2, 3, 4, 5, 6} of the regular hexagon. Give the cycle decomposition of each element. (Number the vertices of the hexagon as in class, clockwise, with 1 at the top. r is rotation clockwise and s is reflection in the vertical axis.)

If we think of groups as a category, as in the Optional Reading, an isomorphism of groups ϕ : G → H is a homomorphism that has an inverse homomorphism (i.e. there is a homomorphism ψ : H → G such that ψϕ = 1_{H} and ϕψ = 1_{G}, where 1_{G} and 1_{H} mean the identity maps of G and H rather than the identity elements of these groups). The existence of ψ certainly implies that ϕ is a bijection. Question 2 shows that in order for a homomorphism ϕ to be an isomorphism it suffices that ϕ be a bijection. The inverse set map is automatically a homomorphism. But in some categories (such as Top, the category of topological spaces and continuous maps) this is not the case. For example the obvious bijection from [0, 1] ⊔ (1, 2] to [0, 2] is continuous but the inverse set map is not continuous. Here [a, b] is the closed interval {x ∈ ℝ|a ≤ x ≤ b, (a, b] is the half open interval {x ∈ ℝ|a < x ≤ b} and ⊔ denotes the disjoint union of topological spaces. With regard to Question 3 note that D_{8} and Q_{8} both contain 8 elements and that D_{24} and S_{4} both contain 24 elements, but these pairs of groups are claimed to be not isomorphic. On the other hand D_{6} and S_{3} both contain 6 elements and are isomorphic (by the second example on page 39). In question 4, ℚ is to be contrasted with ℝ where the corresponding groups are isomorphic (Example 2, page