Math 120A: Extra Questions for Midterm Definitions Complete the following sentences. 1. The direct product of groups G and H is the set under the group operation 2. The symmetric group on n-letters S n is defined as the set of under 3. The dihedral group D n is 4. The orbits of an element σ S n are the sets Questions 1. Show directly (not by quoting a result from the notes) that the subgroup of (Z, +) generated by {4, 6} is 2Z = {2n : n Z}. 2. Which of the following sets generate the group (Z 30, + 30 )? For those sets that do not generate the group give the subgroup that they do generate. (a) {2} (b) {2, 6} (c) {2, 3} (d) {7} (e) {12, 18} (f) {6, 15} 3. List the elements of the following direct product groups: (a) Z 2 Z 3. (b) Z 3 Z 3. (c) Z 2 Z 2 Z 2. 4. Find the orders of the following elements: (a) (1, 3) Z 2 Z 4. (b) (4, 2, 1) Z 6 Z 4 Z 3. (c) ((123), (15)(234)) S 3 S 5. (d) (µ 1, µ 1 ) D 3 D 4 (note the µ 1 s are different elements in different groups here!). 5. Write down the cyclic subgroups generated by each of the elements in the previous question (list all of the elements explicitly). 6. What is the highest possible order of an element of the group S 3 Z 6 V? Exhibit one.
7. Prove that a direct product of Abelian groups is Abelian. 8. Show that the group S 3 is indecomposable: there are no groups G, H of order less than S 3 for which S 3 = G H. (Assuming S 3 is decomposable, there is only one possible decomposition. Why does this decomposition make no sense?) 9. Which of the following maps are permutations? (a) f : Z Z such that f (x) = x 7. (b) f : Z Z such that f (x) = 3x + 4. (c) f : R R such that f (x) = x 3 x. (d) f : R R such that f (x) = x 3 + x. (e) f : {fish, horse, dog, cat} {fish, horse, dog, cat} where fish horse f : horse dog = cat dog. cat fish 10. Let n 3. Prove that if σ S n commutes with every other element of S n (i.e. σρ = ρσ, ρ S n ) then σ is the identity. 11. Give examples of two non-isomorphic non-abelian groups of order 360. 12. Consider the dihedral group D 7 of symmetries of the regular heptagon, viewed as a subgroup of S 7. Each µ i is reflection across the adjacent dashed line whilst ρ j is rotation j steps counterclockwise. ( ) 1 2 3 4 5 6 7 (a) Complete the following: µ 4 =. 7 µ 3 6 (b) Write µ 3 ρ 1 in cycle notation as a product of disjoint cycles. What element of D 7 is µ 3 ρ 1? (c) Argue geometrically that the following rules hold: Rotation Rotation = Rotation. Reflection Reflection = Rotation. Rotation Reflection = Reflection. Hence, or otherwise, calculate (µ 3 ρ 5 ) 666. (e = ρ 0 is considered a rotation; think about what rotations and reflections do to the heptagon) µ 1 4 5 µ 7 µ 2 6 µ 3 ρ 1 = (1234567), ρ j = ρ j 1 13. The set of rotations of a regular octohedron (8 faces, each an equilateral triangle) forms a group under composition. (a) The order of the group can be derermined by choosing a face A. The octohedron can be rotated so that each A is moved to any of the eight faces. This face can now be rotated to any of three positions. What is the order of the group of rotations? 2 7 µ 5 1 µ 4
(b) Repeat your argument with the cube. (c) Imagine placing a dot at the center of each of the triangular faces of a regular octohedron. If you join these dots, what solid do you get? What does this construction tell you about the groups in parts (a) and (b)? (d) The remaining Platonic solids are the tetrahedron (4-faces), the dodecahderon (12-faces), and the icosahedron (20-faces). Find the order of the rotation group of each. Can you make a similar argument to that in part (c) for two of these groups. 14. Find the orbits of the following permutations: (a) (145)(2345) S 5. (b) (154)(254)(1234) S 5. (c) (1574)(324)(3256) S 7. (d) σ : Z Z where σ(n) = n + 3. 15. What are the orders of each of the permutations in the previous question? 16. What is the maximum order of an element in each of the groups S 4, S 5, S 6, S 7, S 8? Exhibit such an element in each case. 17. For what integers n does the subgroup relation C n S 8 hold? Explain your answer. 18. In this question we work in the group S 9. Let σ and τ be the following permutations: σ = ( 1 2 3 4 5 6 7 8 ) 9 3 7 5 9 1 2 4 8 6 (a) Complete the following ( ) 1 2 3 4 5 6 7 8 9 τσ =. 2 and τ = (1532)(69). (b) Calculate στ in cycle notation, making sure you simplify to disjoint cycles. (c) What is the order of σ? (d) Compute (στ) 432 σ 43 as a product of disjoint cycles. (e) Using your answer to (c), construct the entire subgroup diagram of σ and give a generator for each subgroup. Optional: for a challenge... 19. Prove that D n is a subgroup of A n n 1 (mod 4) (If you are happy doing this in one shot then do so, otherwise use the following steps to guide your thinking). (a) Label the corners of a regular n-gon 1 through n counter-clockwise so that every element of D n may be written as a permutation of {1, 2,..., n}. Write in a sentence what you are required to prove: i.e. what is the condition that characterises being in the group A n?
(b) Consider the rotation ρ 1 = (123 n) of the n-gon one step counter-clockwise. Is ρ 1 odd or even, and how does this depend on n? (c) Show that every rotation ρ i D n is generated by ρ 1. When is the set of rotations in D n a subgroup of A n? (d) Consider a reflection µ D n. µ permutes corners of the n-gon by swapping pairs. How many pairs of corners does µ swap when n 1 (mod 4)? Is µ an odd or even permutation? You may use a picture, provided it is sufficiently general. (e) Summarize parts (a d) to argue the direction of the theorem. (f) Prove the direction of the theorem by exhibiting an element of D n which is not in A n for every case when n 1 (mod 4). 20. This question considers homomorphisms between finite cyclic groups. You should be completely comfortable with parts (a f). Parts (g j) are beyond the level of the course. (a) Suppose that ψ : (Z m, + m ) (Z n, + n ) is a homomorphism, and suppose that ψ(1) = k. Prove that x Z m, ψ(x) = kx (mod n) (b) By part (a) there are at most 24 distinct homomorphisms ψ : Z 36 Z 24 : each must be of the form ψ k : x kx (mod 24) for some k Z 24. However, not all choices of k actually define a function. Suppose that k = 3. Show that ψ 3 (37) = ψ 3 (1) Why does this show that ψ 3 is not well-defined? (c) Prove that ψ k : Z 36 Z 24 : x kx (mod 24) is well-defined if and only if k 0 (mod 2). How many distinct homomorphisms are there ψ k : Z 36 Z 24? (d) Are any of the homomorphisms ψ k : Z 36 Z 24 injective? Surjective? (e) Find all of the homomorphisms ψ : Z 12 Z 24. Are any injective or surjective? (f) Prove that ψ k : Z m Z n is a well-defined homomorphism iff k 0 (mod n d ) where d = gcd(m, n). How many distinct homomorphisms are there? (g) Prove the following: i. There exists an injective homomorphism ψ : Z m Z n if and only if m n. ii. There exists a surjective homomorphism ψ : Z m Z n if and only if n m. Hints: The image of a homomorphism ψ is a subgroup of Z n. What is the order of the image if ψ is 1 1? What does it mean if 1 Z n is in the image of ψ? (h) Prove that the number of distinct isomorphisms ψ : Z n Z n is given by the Euler totient function: ϕ(n) = Z n = {x Z n : gcd(x, n) = 1} which returns the number of units in Z n under multiplication. Hint: If ψ is an isomorphism, what can ψ(1) be?
(i) Prove that the set of units Z n forms a group under multiplication. Hint: Recall the Euclidean Algorithm: gcd(x, n) = 1... (j) The set of isomorphisms of Z n with itself forms a group under composition, the automorphism group Aut(Z n ). Prove that Aut(Z n ) = (Z n, ). Hint: Re-read the proof of Cayley s Theorem.