School of Science
(Mathematical Sciences)
MATH2148/MATH2172 Assignment 1
(Worth 25% of final mark)
DUE: 11:59 pm Saturday 1st May 2021
FULL WORKING MUST BE SHOWN.
1. Compute your special integer n as follows.
• 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
(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
(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)
1
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. Use your special integer n to compute your special index number i as follows.
• Divide n by 22 and find the remainder.
• Add 1 to the remainder, and call the resulting number i.
Locate matrix number i in the list on the next page. This is your generator matrix G 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) Find a received word r1 which is a Hamming distance of 1 away from a codeword. Is r1
(e) Find a non-correctible received word r2. Explain why r2 can’t be corrected.
(f) Find a correctible received word r3 which is a Hamming distance of 2 away from the
nearest codeword. Explain why r3 can be corrected.
(20 + 5 + 5 + 5 + 5 + 5 = 45 marks)
2
List of Matrices
1.
 1 0 0 1 1 1 10 1 0 1 1 0 0
0 0 1 0 0 1 1
 2.
 1 0 0 1 1 1 10 1 0 0 0 1 1
0 0 1 1 1 0 0
 3.
 1 0 0 1 1 1 10 1 0 1 0 1 0
0 0 1 0 1 0 1

4.
 1 0 0 1 1 1 10 1 0 1 0 0 1
0 0 1 0 1 1 0
 5.
 1 0 0 1 1 1 10 1 0 0 1 1 0
0 0 1 1 0 0 1
 6.
 1 0 0 1 1 1 10 1 0 0 1 0 1
0 0 1 1 0 1 0

7.
 1 0 0 1 1 1 00 1 0 0 1 0 1
0 0 1 0 0 1 1
 8.
 1 0 0 1 1 1 00 1 0 0 0 1 1
0 0 1 0 1 0 1
 9.
 1 0 0 1 1 1 00 1 0 0 1 0 1
0 0 1 1 0 0 1

10.
 1 0 0 1 1 1 00 1 0 1 0 0 1
0 0 1 0 1 0 1
 11.
 1 0 0 1 1 0 10 1 0 0 0 1 1
0 0 1 0 1 1 0
 12.
 1 0 0 1 1 0 10 1 0 0 1 1 0
0 0 1 0 0 1 1

13.
 1 0 0 1 1 0 10 1 0 1 0 1 0
0 0 1 0 1 1 0
 14.
 1 0 0 1 1 0 10 1 0 0 1 1 0
0 0 1 1 0 1 0
 15.
 1 0 0 1 0 1 10 1 0 0 1 0 1
0 0 1 0 1 1 0

16.
 1 0 0 1 0 1 10 1 0 0 1 1 0
0 0 1 0 1 0 1
 17.
 1 0 0 1 0 1 10 1 0 1 1 0 0
0 0 1 0 1 1 0
 18.
 1 0 0 1 0 1 10 1 0 0 1 1 0
0 0 1 1 1 0 0

19.
 1 0 0 0 1 1 10 1 0 1 0 0 1
0 0 1 1 0 1 0
 20.
 1 0 0 0 1 1 10 1 0 1 0 1 0
0 0 1 1 0 0 1
 21.
 1 0 0 0 1 1 10 1 0 1 1 0 0
0 0 1 1 0 1 0

22.
 1 0 0 0 1 1 10 1 0 1 0 1 0
0 0 1 1 1 0 0

3 