exam-May-fulldraft2022.pdf

King’s College London University Of London This paper is part of an examination of the College counting towards the award of a degree. Examinations are governed by the College Regulations under the authority of the Academic Board. PLACE this paper and any answer booklets in the EXAM ENVELOPE provide...

King’s College London University Of London This paper is part of an examination of the College counting towards the award of a degree. Examinations are governed by the College Regulations under the authority of the Academic Board. PLACE this paper and any answer booklets in the EXAM ENVELOPE provided Candidate No:................................. Desk No:........................ BSc Examination 4CCM121A/5CCM121B Introduction to Abstract Algebra Summer 2023 Time Allowed: Two Hours This paper consists of two sections, Section A and Section B. Section A contributes half the total marks for the paper. Answer all questions. NO calculators are permitted. DO NOT REMOVE THIS PAPER FROM THE EXAMINATION ROOM TURN OVER WHEN INSTRUCTED 2023 ©King’s College London Section A 4CCM121A/5CCM121B SECTION A Fill in your answers on the ANSWER SHEET for Section A. You will find the answer sheet at the back page of the exam paper. In Section A there are 5 marks for a correct answer, -1 point for an incorrect answer, and 0 points for a blank answer. A 1. Each of the following options gives information about two integers a and b. Choose the option in which gcd(a, b) > 1. A) There exist m, n ∈ Z, not both zero and with greatest common divisor d, such that a = md , and b = nd. B) For any prime number p, if p | a, then p ∤ b. C) Let S = {ax + by | x, y ∈ Z}. Then S ̸= Z. D) Za = ⟨[b]a ⟩. A 2. Each of the following options gives a set S and a binary operation ∗ on S. Select the option in which ∗ is both associative and commutative. A) S = R and x ∗ y = xy + 1 for all x, y ∈ R. B) S is the set of functions f : Z → Z, and f ∗ g = f ◦ g for all f, g ∈ S. C) S = Z × Z and (a, b) ∗ (c, d) = (ad + bc, bd) for all (a, b), (c, d) ∈ S. D) S = S3 , and ∗ is given by f ∗ g = f gf −1 for all f, g ∈ S3. -2- See Next Page Section A A 3. 4CCM121A/5CCM121B Each of the following options gives information about a group G and two elements g, h ∈ G. Select the option that does not imply h = g −1. A) ϕ : G → Z5 is a group homomorphism, and ϕ(gh) = 5. B) ϕ : Z5 → G is a group homomorphism, g = ϕ(5 ), and h = ϕ(5 ). C) G = S5 , g = (1 3 2 5), and h = (1 5 2 3). D) G = D5 , g is a rotation in G, and h = g 4. A 4. Each of the following options gives information about a group G and an element g ∈ G. Select the option in which g has order greater than 3. A) G = S5 and g = (12)(13)(24). B) ϕ : Z6 → G is a group homomorphism, and g = ϕ(6 ). C) G is any group, and g = g 4. D) G = D4 , and g is conjugate to a reflection. A 5. Each of the following options gives a group G and a subset H of G. In each case, we write e for the identity element of G. Select the option in which H forms a subgroup of G. A) G = S3 and H = {e, (12), (13), (23)}. B) G = Z, n ∈ Z with n > 1, and H is the congruence class of 1 modulo n. C) G = M2 (R) and H = GL2 (R). D) G is any group, g ∈ G is an element of order 2, and H = {e, g}. -3- See Next Page Section A A 6. 4CCM121A/5CCM121B Each of the following options gives information about a group G. Choose the option for which G is not cyclic. A) G = Z8. B) G = Z× 5. C) G is a nonabelian group. D) There exists a surjective homomorphism ϕ : Z → G. A 7. Each of the following options gives information about two groups G and H, and about a function ϕ : G → H. Select the option in which ϕ does not define a group homomorphism. A) G = S3 and H = S4. If f ∈ S3 , define ϕ(f ) : {1,... , 4} → {1,... , 4} by ( f (k) if k ∈ {1, 2, 3} ϕ(f )(k) = 4 if k = 4. B) G = Z2 , H = Z6 , and ϕ is defined by ϕ(2 ) = 6 , ϕ(2 ) = 6. C) G = R, H = GL2 (R), and ϕ is defined by ϕ(x) = ( 10 x1 ) for all x ∈ R. D) G = Z, H is any group, h ∈ H, and ϕ(n) = hn for all n ∈ Z. A 8. Each of the following options gives two groups, G and H. Choose the option in which G is not isomorphic to H. A) G = S3 and H = D3. B) G = Z2 × Z2 and H = Z× 8. C) Let n, m ∈ N with gcd(n, m) = 1. Let G = Zn × Zm and H = Znm. D) Let n ∈ Z with n ≥ 2. Let G = Zn × Zn and H = Zn2. -4- See Next Page Section A A 9. 4CCM121A/5CCM121B Each of the following options gives a set S and a relation ∼ on S. Choose the option in which ∼ defines an equivalence relation. A) S = Z and n ∼ m if n | m. B) S is a group G, and for all x, y ∈ G, x ∼ y if there exists g ∈ G such that gx = yg. C) S = R and x ∼ y if |x − y| < 1. D) S is the set of all groups, and G ∼ H if there exists an injective homomorphism ϕ : G → H. A 10. Each of the following options gives information about a ring R and an element r ∈ R. Select the option in which r is not a unit in R. A) R = R[x] and r = x. B) R = M2 (R) and r = ( 01 10 ). C) R = Z6 and r = 6. D) R is any ring, s, t ∈ R are units, and r = st. -5- See Next Page Section B 4CCM121A/5CCM121B SECTION B Answer all questions. Unless otherwise noted, each solution needs to contain enough intermediate steps to indicate how to reach the final answer. Question B10 is worth 16 points. Questions B11 and B12 are worth 17 points each. B 11. Let d = gcd(2223, 111) and let M = 246753 = 111 · 2223. a) Use the Euclidean algorithm to find d and to find a, b ∈ Z such that d = 2223a + 111b. Clearly label your answers for a, b, and d. b) Let n, k, x, y ∈ Z such that k | n and x ≡ y mod n. Show x ≡ y mod k. [For full credit, you must prove any results you use.] c) Describe all integers x such that x ≡ 1 mod 2223 and x ≡ 14 mod 111. Explain your reasoning. d) Let ϕ : ZM → Z2223 × Z111 be defined by ϕ([n]M ) = ([n]2223 , [n]111 ). Find ker ϕ and |ker ϕ|. (You do not need to prove that ϕ is a homomorphism.) B 12. Let G be a group and let H be a subgroup of G. Let g ∈ G. a) Write the definition of the left coset gH. b) Show that gH is a subgroup of G if and only if g ∈ H. [You may use standard facts about subgroups, but you must prove any results you use about cosets.] c) Suppose H ∪ gH is a subgroup of G. Show that ord(g 2 ) divides |H|. [You may use any result from the course, but you should give the full statement of any result you use.] B 13. √ Let R = {a + b 2 | a, b ∈ Z}. a) Show that R is a subring of R. b) Show that R and Z × Z are not isomorphic rings. [For full credit, you must prove any result that you use.] c) Is R a field? Explain why or why not. d) Show that there exists a ring homomorphism ϕ : R → Z2. -6- See Next Page Section B 4CCM121A/5CCM121B This page intentionally left blank. -7- See Next Page Section A SECTION A – ANSWER SHEET 4CCM121A/5CCM121B Candidate No. Each question has exactly one correct answer. The marks are +5 for the correct answer, −1 for a wrong answer, and 0 marks for no answer or if more than one answer is chosen. If the resulting total is negative, your final score on Section A will be rounded up to 0. Indicate your chosen answer to each multiple-choice question with a tick in the appropriate box. If you want to change an answer, please fill in the box that contains the unintended tick and write your final answer in the right-hand column: Question 1 A ✔ B C D E Question 2 A ✔ B C D E B Only answers recorded on the solution grid or adjacent column will be counted. You can use separate booklets for rough work, but this will not be marked. Question A1 A B C D E Question A2 A B C D E Question A3 A B C D E Question A4 A B C D E Question A5 A B C D E Question A6 A B C D E Question A7 A B C D E Question A8 A B C D E Question A9 A B C D E Question A10 A B C D E -8- Final Page