- May 15, 2020

MATH1061 Assignment 4 Semester 2/2019 This Assignment is compulsory, and contributes 5% towards your final grade. It must be submitted by 10am on Friday 25 October, 2019. In the absence of a medical certificate or other valid documented excuse, assignments submitted after the due date will not be marked. Submission: You will receive a coversheet for this assignment by email. Print that coversheet, staple it to the front of your assignment (which may be handwritten), and submit your assignment using the assignment submission system in the corridor between buildings 69 and 62. Q1 Consider the group (Q × Q,+), where Q × Q = {(a, b) | a, b ∈ Q}, and where addition is defined in the usual way by (a, b) + (c, d) = (a+ c, b+ d). So, for instance, (23 ,−12) ∈ Q×Q, and (12 ,−34) + (14 , 1) = (34 , 14). (a) What is the identity in this group? You do not need to justify your answer. (b) What is the inverse of the element (x, y) ∈ Q × Q? You do not need to justify your answer. (c) Is the group (Q×Q,+) cyclic? Explain why / why not. (d) Is the group (Q×Q,+) isomorphic to the group (Q,+)? Explain why / why not. (10 marks) Q2 (a) Write out the Cayley tables for (Z4,+) and (Z4, ·). (b) Explain why (Z4,+, ·) is not a field. (c) Create a field with exactly four elements. You should give your answer using two Cayley tables: one for the addition operation, and one for the multiplication operation. You do not need to justify why this is a field. (10 marks) Q3 Consider the group (Z3 × Z3,+), where again Z3 × Z3 = {(a, b) | a, b ∈ Z3}, and we define addition by (a, b) + (c, d) = (a + c, b + d). So, for instance, (1, 2) + (0, 2) = (1, 1). (a) List all of the subgroups of (Z3 × Z3,+). You should explain why your list is complete (i.e., why there are no subgroups other than the ones you have written). (b) Which of these subgroups are cyclic? You do not need to justify your answers. (10 marks) Q4 A university has a 14-day exam period. You need to choose three days from this period: one for your maths exam, one for your physics exam, and one for your linguistics exam. Unless otherwise specified, it is allowed for multiple exams to be held on the same day. Here order does matter: for instance, if you put maths, physics and linguistics on days 5, 6 and 10 respectively, this is a different choice from putting maths, physics and linguistics on days 6, 10 and 5 respectively. (a) How many choices of days are possible? (b) How many choices of days are possible if you are not allowed to hold all three exams on the same day? (c) How many choices of days are possible if all three exams must be held on three different days? (d) How many choices of days are possible if the maths exam must be held at least two days earlier than physics, and the physics exam must be held at least two days earlier than linguistics? (So, for example, you could put maths, physics and linguistics on days 2, 4 and 7 respectively, but not days 3, 4 and 8 respectively.) (10 marks)