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SECTION A
- Prove by induction that, for every integer n ≥ 2,
∑^ n
k=
k^2 − 1
2 n + 1 2 n(n + 1)
[6 marks]
- Find the greatest common divisor d of 1870 and 4641, and find integers s and t such that d = 1870s + 4641t. [6 marks]
- Find the smallest positive integer x that satisfies the congruences
x ≡ 14 mod 37, x ≡ 5 mod 42.
Find also the next integer that satisfies these congruences. [6 marks]
- In each of the following cases find the solutions (if any) of the given linear congruence:
(a) 8 x ≡ 14 (mod 36); (b) 8 x ≡ 12 (mod 36); (c) 8 x ≡ 12 (mod 37). [10 marks]
- (a) Let f : Z 7 → Z 7 be given by f (x) = x^3 + x; draw the diagram of f and determine whether f is surjective and whether it is injective. Justify your answer.
(b) Give an example of a three element subset A ⊂ Z 7 such that the restriction of f to A is injective. Justify your answer.
(c) Determine the image of f. Justify your answer.
[9 marks]
- Let π, ρ be the permutations
π =
, ρ = (1356)(24376).
Write π, ρ, ρ^2 and πρ as products of disjoint cycles and determine their orders and signs. [8 marks]
- List the elements in the group G 16 of invertible congruence classes mod- ulo 16; construct a multiplication table for G 16. List the elements of order 2 in this group. In the multiplication table you can write k for [k] 16. [10 marks]
SECTION B
- State the axioms for a group. In each of the following cases, determine whether ∗ defines a binary operation on H, and if so, which of the group axioms are satisfied. Justify your answers. (A non-justified or incorrectly justified answer is worth half the points if all the answers in the exercise are correct and is worth zero otherwise.) [You may assume that composition of maps and multiplication modulo n are associative.]
(a) H is the set of positive real numbers under substraction; (b) H = Z 11 , with ∗ given by multiplication in Z 11 ; (c) H is formed by the congruence classes of 3,6,9 and 12 modulo 15, with ∗ given by multiplication in Z 15. [15 marks]
Recall that we denote the group of all invertible congruence classes modulo n ≥ 2 by Gn.
(a) Say what it means for a subset H of a group G to be a subgroup of G. Let H = {[1] 9 , [4] 9 , [7] 9 }. By constructing a multiplication table for H or otherwise, show that H is a subgroup of G 9. [3 marks]
(b) Say what it means for a group G to be cyclic. Is G 9 cyclic? If it is, give a generator of this group. Is the group H from question 9 (a) cyclic? If it is, give a generator of this group. [7 marks]
(c) Say what it means for a subset of a group G to be a right coset. List all right cosets of H from question 9 (a) in G 9. [5 marks]
Let G be a group and let H be a subgroup of G. (i) Give the definition of a right coset of H in G. [3 marks] (ii) From now on we assume in this exercise that G is finite. State Lagrange’s theorem. [2 marks]
(iii) Prove that any element of G belongs to some right coset. [1 marks] (iv) Prove that any two right cosets coincide or do not intersect. [3 marks] (v) Prove that any two right cosets of H contain the same number of elements. [3 marks] (vi) Deduce Lagrange’s theorem from (iii), (iv) and (v). [3 marks]
