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SECTION A
- Let f be the map with the following diagram.
Determine whether it is injective and whether it is surjective. Explain your answers. Let A′^ = { 2 , 3 , 5 }, B′^ = { 1 , 3 , 4 }. Determine the image f (A′) of A′^ and the preimage f −^1 (B′) of B′. Draw the diagram of the restriction f |A′ of f to A′. Determine whether it is injective and whether it is surjective and explain the answers.
[11 marks]
- In each of the following cases find the solutions (if any) of the given linear congruence:
(a) 16 x ≡ 20 (mod 37); (b) 16 x ≡ 20 (mod 36); (c) 15 x ≡ 19 (mod 35). [10 marks]
- Solve the simultaneous congruences
x ≡ 4 (mod 15), x ≡ 11 (mod 17), x ≡ 6 (mod 14)
expressing your answer in the form x ≡ a (mod n) for suitable a and n. Find the least positive integer that satisfies these congruences.
[10 marks]
- Let π, ρ be the permutations
π =
, ρ = (43752)(1527).
Write π, ρ, ρ^2 and πρ as products of disjoint cycles and determine their orders and signs. [8 marks]
Paper Code MATH 142 Page 2 of 4 CONTINUED
- List the elements in the group G 30 of invertible congruence classes mod- ulo 30; construct a multiplication table for G 30. List the elements of order 2 in this group. In the multiplication table you can write k for [k] 30.
[10 marks]
- (i) Give the definition of Euler’s function n 7 → ϕ(n), n ∈ Z, n ≥ 1. (ii) State a formula for ϕ(p^3 ), where p is a prime. Find ϕ(125). (iii) State Euler’s theorem. Use it to determine the remainder when 4^104 + 6302 is divided by 125. Explain why Euler’s theorem is applicable here. [6 marks]
SECTION B
- Give the definition of a binary operation on a set and 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. [You may assume that addition and multiplication of integers and addition and multiplication modulo n are associative.]
(a) H is the set of all congruence classes modulo 7, with ∗ given by the multiplication of congruence classes;
(b) H is the set of all non-zero congruence classes modulo 7 (i.e., the set of those congruence classes that are not equal [0] 7 ), with ∗ given by the multiplication of congruence classes;
(c) H is the set of all non-empty finite subsets of Z with A∗B for A, B ∈ H defined as the set formed by all sums a + b where a ∈ A, b ∈ B. [15 marks]
- (a) Say what it means for a subset H of a group G to be a subgroup of G. [2 marks]
(b) Let H = {[1] 17 , [4] 17 , [13] 17 , [16] 17 }. By constructing a multiplication table for H or otherwise, show that H is a subgroup of G 17.
[4 marks] (c) Say what it means for a group G to be cyclic. Is the group H from question 8 (b) cyclic? [3 marks]
(d) State the group homomorphism theorem. Assuming that every sub- group of Z is cyclic, show that any cyclic group is isomorphic to Z or Zn, n ≥ 1.
[6 marks]
Paper Code MATH 142 Page 3 of 4 CONTINUED
