Need help with the second half of question 1. I do not need help with question 5.
Math 361 - Homework 4 Fall 2021 Instructor: Curtis Wendlandt Due: Tuesday, October 26th at 11:59pm Instructions (1) Please submit complete solutions to the below problems via Canvas by the above indicated due date. Generally speaking, unexcused late homework will not be accepted. (2) Your submission should be a single pdf document, with all questions clearly marked. All writing must be clearly legible; see the Coursework submissions section of the syllabus. (3) For notation and terminology, you should follow the conventions established in class. Centralizers, normalizers and centers Problem 1. 1.1. Let (a0 a1 · · · am) ∈ Sn be an arbitrary m-cycle. Prove that γ(a1 a2 · · · am)γ−1 = (γ(a1) γ(a2) · · · γ(am)) Hint. Any element of Sn is determined by the values it takes on the set X = {1, . . . , n}. What happens if you apply the left-hand side to an element of X which takes the form γ(aj) for some j? What about an element k ∈ X which is not of this form (i.e. which belongs to X \ {γ(a1), . . . , γ(am)})? 1.2. Write down a complete list of elements of the symmetric group S3, expressed in terms of their cycle decompositions. Afterwards, find the normalizers NS3(A) and centralizers CS3(A) of the following sets: a) A = {(1 2), (2 3)} b) A = {(1 2), (2 3), (1 3)} Problem 2. In this question you will compute the center of the n-th dihedral group. 2.1. Prove that if x = rks for some 0 ≤ k ≤ n − 1, then rx ̸= xr. Show that this implies that Z(Dn) ⊂ ⟨r⟩. 1 2 2.2. Show that for 1 ≤ k ≤ n−1, rk ∈ Z(Dn) if and only if n is even and k = n/2. Conclude that Z(Dn) = { {e} if n is odd {e, r n2 } if n is even. Cyclic groups, subgroups generated by subsets, and lattices Problem 3. Let G be an arbitrary finite group. 3.1. Suppose there is an element x ∈ G of order |G|. Show that G is cyclic. 3.2. Suppose again that G is a finite group. Show that the order |x| of any element x ∈ G must divide |G|, and conclude that x|G| = e. Hint. See Problem 5 of Homework 3. 3.3. Suppose now that |G| = p is prime. Determine if G is cyclic. Problem 4. 4.1. Let G and H be two groups, and suppose there is an isomorphism φ : G→ H. Show that G is cyclic if and only if H is cyclic. 4.2. Prove that the following groups are not isomorphic: a) Z× Z2 and Z. b) Z/16Z and D8. Problem 5. Find all distinct subgroups of Z/48Z, and write down all their elements (you can skip writing down all the elements of the full group Z/48Z itself). Use this information to draw the lattice of subgroups of Z/48Z. Problem 6. A group G is said to be finitely generated if there is a finite subset A = {x1, . . . , xn} ⊂ G such that G = ⟨A⟩ = ⟨x1, . . . xn⟩. 6.1. Suppose that H is a finitely generated subgroup of the additive group Q of rational numbers. Prove that H is a cyclic group. Hint. Since H is finitely generated, we can write H = ⟨x1, . . . , xn⟩ for some n > 0 and x1, . . . , xn ∈ Q. Write xi = ai/bi, and let b = b1b2 · · · bn. Show that H ≤ 〈 1 b 〉 and then explain why this implies H is cyclic. 6.2. Show that Q is not itself finitely generated. 3 Problem 7. 7.1. Let G be a cyclic group with |G| = n, and let k ∈ Z be relatively prime to n. Consider the function ψ : G→ G defined by ψ(x) = xk ∀ x ∈ G. Show that ψ is surjective. (Bézout’s Identity might help, but it is not neces- sary) 7.2. Suppose now that G is any finite group G of order n. Prove that the map ψ defined above is still surjective. Hint. For each x ∈ G, consider the map ψx : ⟨x⟩ → ⟨x⟩ defined by ψx(y) = yk for each y ∈ ⟨x⟩. Try to apply 7.1 to show each map ψx is surjective. Then prove that this implies that ψ is surjective. Instructions Centralizers, normalizers and centers Problem 1 Problem 2 Cyclic groups, subgroups generated by subsets, and lattices Problem 3 Problem 4 Problem 5 Problem 6 Problem 7