MTH 122 Introduction to Abstract Algebra 2018/19
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MTH 122
2nd SEMESTER 2018/19 RESIT EXAMINATION
Introduction to Abstract Algebra
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Q 1. Consider the following permutations in S9 :
a- = |
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|
2 9 |
3 4 |
4 8 |
5 1 |
6 6 |
7 3 |
7(8) ;) |
(a) Express a- and r as a product of disjoint cycles.
(b) Compute the order of a- and of T (explaining your reasoning).
(c) Compute ro- and o-r.
[15 marks]
Q 2. Assume that ¢: G -+ His a group homomorphism.
(a) Assume that g E G has finite order. Prove that the order of cp(g) divides the order of g. (b) Give the definition of ker(¢).
(c) Prove that ¢ is injective if and only if ker(c/>) is trivial.
(d) Assume !GI = 84 and IHI = 17. Prove that ¢ is trivial.
(e) Assume IGI = IHI = 31. Prove that ¢ is either trivial or an isomorphism.
[25 marks]
Q 3. (a) Give the definition of a ring.
(b) For each of the following, say if it is a ring or not:
i. N, with the usual product and sum;
ii. the set Mat2 x 3 (IR) of all 2 x 3 matrices with entries in IR, with the usual matrix operations;
iii. the set Z[x] of all polynomials with coefficients in Z, with the usual product and sum of polynomials. If you say it is not a ring, provide a reason.
[15 marks]
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Q 4. Let R be a commutative ring with identity (that is, ab= ba for every a, b E R and there is an element 1 E R,
(a) Prove that if xE R is a unit, then xis not a zero-divisor and vice versa, if xis a zero-divisor then xis not a unit.
(b) Prove that Z/52Z contains some zero-divisor.
(c) Give the definition of a field.
(d) Prove that if n is not prime, then Z/nZ is not a field.
(12 marks)
Q 5. For each of the following equations, find a solution x E Z or prove that no solution x E Z exists:
In all cases, explain your reasoning. [15 marks]
Q 6. Let V C F1[x] be the set of polynomials of degree at most 4. Let ¢: V -----+ F
be the map ¢(!) := (f(O),f(l), f(2), f(3), f(4),f(5),f(6)) .
Put C := ¢(V).
(a) Prove that ¢ is a linear map. (b) Prove that ¢ is injective.
(c) Prove that C is a linear code.
(d) Compute the minimum distance of C.
(e) How many errors in a code word x= (x1,x2,x3,x4,x5,x5,x7) EC can reliably be detected?
[18 marks]
2022-08-01