I' rushing to finish my other assignment, please help me do as much as possible.
School of Science (Mathematical Sciences) MATH2148/MATH2172 Assignment 1 (Worth 18% of final mark) DUE: 11:59 pm Saturday 25th April 2020 FULL WORKING MUST BE SHOWN. 1. Compute the positive integer n as follows. • Add up the digits in your student identity number. • Call this number n unless it is prime, in which case add 1 to it and call the answer n. (a) We consider all possible groups of order n. (i) How many such groups are there? [Quote your source.] (ii) How many of those groups are abelian? [Quote your source.] (iii) For a group G of order n, what numbers might possibly occur as the orders of the elements of G? Can we be sure that any of those numbers will definitely occur as the order of an element of the group? [Justify your answer.] (iv) For a group G of order n, what numbers might possibly occur as the orders of the subgroups of G? Can we be sure that any of those numbers will definitely occur as the order of a subgroup of the group? [Justify your answer.] (b) Consider the symmetric group Sn of all permutations of the set X = {1, 2, 3, . . . , n}. (i) What is the order of Sn? (ii) How many of the permutations represent symmetries of a regular n-gon? (iii) Describe all the symmetries of a regular n-gon. For those that are rotations, specify the angle of rotation in each case. For those that are reflections, specify the axis about which each reflection takes place. ((5 + 5 + 10 + 10) + (5 + 5 + 10) = 50 marks) 2. Let X = {1, 2, 3, 4}. (a) Write down all the permutations of X as (2× 4)-matrices. (b) Which of these permutations are symmetries of the square with vertices 1, 2, 3 and 4? For each one that is a symmetry, give it the name ρi (for a suitable choice of i) if it is a rotation and σj (for a suitable choice of j) if it is a reflection. (c) Using your answers to part (b), construct the Cayley table for the dihedral group D4. (d) Find all subgroups of D4. (4× 10 = 40 marks) 3. Let G = {u, v, w, x, y, z} be the 6-element group that has been assigned to you. (i) Find the identity element e of G. (ii) Find the inverse of each element of G. (iii) For each element of G, find the cyclic subgroup that it generates. (iv) Find all the subgroups of G. (v) Choose an element of order 3 in G, and call it a. Find all the distinct cosets of H = 〈a〉 in G, and write out the multiplication table for the quotient group G/H. (vi) Choose an element of order 2 in G, and call it b. Let Z5 = {0, 1, 2, 3, 4} be the 5-element cyclic group under addition modulo 5. Let (K, ∗) be the direct product of Z5 and 〈b〉. Construct the Cayley table for (K, ∗). (5 + 5 + 10 + 10 + 10 + 10 = 50 marks) 4. Let G = 1 0 0 1 0 0 10 1 0 0 1 0 1 0 0 1 1 0 1 1 be the generator matrix for a group code C. (a) Construct an encoding table for C. As well as a column for the message words and a column for the code words, include a column for the weights of the code words. (b) Determine the parameters [n, k, d] of the linear code C. (c) What are the error-detecting and error-correcting capabilities of the code? (d) You receive the word 0101011. Can this be corrected? Explain your answer. (e) You receive the word 1110100. Can this be corrected? Explain your answer. (20 + 5 + 5 + 5 + 5 = 40 marks)