MAS2006

UNIVERSITY OF EXETER SCHOOL OF MATHEMATICAL SCIENCES

GROUPS, RINGS AND FIELDS

January 2002  9:30 a.m. – 12:30 p.m.

Duration: 3 hours

Examiner: Robin Chapman

Answer Section A (40%) and any THREE of the four questions in Section B (20% for each). Marks shown in questions are merely a guideline. Calculators labelled as approved by the School of Mathematical Sciences may be used.

SECTION A

1. (a) Draw a regular hexagon (six sides). Describe each of its symmetry operations, and give the order of each. Find all the cyclic subgroups of the group of symmetry operations of the hexagon. (12)

(b) Calculate the orders and signs of the following permutations:

(i) (1 4 8)(2 7)(3 5); (ii) (1 10 7)(2 9 5 4 6 8 3); (iii) (1 11 3)(2 12 9 8 4 7 10)(5 6).

(3)

(c) Find the units in the ring Z_{16} of integers modulo 16. For each unit find its inverse and its order in the group of units. Show that 8 + 3\sqrt{7} is a unit in the ring

R = Z[\sqrt{7}] = \{a + b\sqrt{7} : a, b ∈ Z\},

and hence deduce that R has infinitely many units. (6)

(d) Recall that the conjugate of a quaternion α = a + bi + cj + dk (where a, b, c, d ∈ R) is \overline{α} = a−bi−cj−dk, and that a quaternion α is pure if a = 0. Show that α is pure if and only if α^{2} is real and α^{2} ≤ 0. Let α be a pure quaternion and β be any quaternion. Show that βα\overline{β} is a pure quaternion. (You may use, without proof, the fact that β\overline{β} = \overline{β}β is a nonnegative real number.) (6)

(e) Let

R = M_{2}(R) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) : a, b, c, d ∈ R\} .

Which of the following sets are subrings of R? Justify your answers.

(i)

S_{1} = \{( \begin{matrix}a & b \\ −b & a \end{matrix} ) : a, b ∈ R\};

(ii)

S_{2} = \{( \begin{matrix}a & b \\ 0 & b \end{matrix} ) : a, b ∈ R\} .

(7)

(f) In each of the following cases use the Euclidean algorithm to find g = gcd(a, b). Also find r, s ∈ Z with g = ra + sb.

(i) a = 74, b = 54; (ii) a = 199, b = 123;

(iii) a = 2001, b = 1701. (6)

[40]

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SECTION B

2. (a) Give the definitions of the terms group and subgroup. Show that if H_{1} and H_{2} are subgroups of a group G then H_{1} ∩ H_{2} is a subgroup of G. (8)

(b) In each case determine whether the set H is a subgroup of the group G. Justify your answers.

(i) G = C^{∗} with the operation of multiplication, H = {z ∈ C^{∗} : |z| ≥ 1};

(ii) G = Q^{∗} with the operation of multiplication,

H = {m/n : m, n ∈ Z, m ≡ n ≡ 1 (mod 10)}; (iii) G = C^{∗} with the operation of multiplication,

H = {x + iy ∈ C^{∗} : x, y ∈ R, xy = 0}. (12)

[20]

3. (a) Let H be a subgroup of a group G. What is a left coset aH of H in G? What is a right coset? Show that if aH and bH are left cosets of H in G then either aH = bH or aH ∩ bH = ∅. If H has exactly two left cosets in G, namely H and aH, show that aH is also a right coset of H in G. (10)

(b) Find the left cosets of the following subgroup of S_{4}:

H = {id, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}.

(6)

(c) Let G = S_{3}. Find a subgroup H of G and a left coset of H which

is not a right coset of H. (4) [20]

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4. (a) Let G and H be groups. What is a homomorphism from G to H? What is the kernel of a homomorphism? Let K be the kernel of the homomorphism ϕ : G → H. Show that K is a subgroup of G and that Ka = aK for each a ∈ G. [You may assume that ϕ(e) = e′ where e and e′ denote the identities of G and H respectively, and that ϕ(a^{−1}) = ϕ(a)^{−1} for each a ∈ G.] (10)

(b) In each of the following cases determine whether ϕ : G → H is a homomorphism. Whenever ϕ is a homomorphism, find its kernel. Justify your answers.

(i) G = C with the operation of addition, H = C^{∗} with the operation of multiplication, ϕ(z) = exp(πiz).

(ii) G = H = GL_{2}(R) with the operation of multiplication,

ϕ (( \begin{matrix}a & b \\ c & d \end{matrix} )) = ( \begin{matrix}a & c \\ b & d \end{matrix} ) .

(iii) G = R with the operation of addition, H = GL_{3}(R) with the operation of multiplication,

ϕ(x) = ( \begin{matrix}1 & 0 & x \\ 2x & 1 & 2x + x^{2} \\ 0 & 0 & 1 \end{matrix} ) .

(10) [20]

5. (a) Give the definition of the term Euclidean domain. Let R be a Euclidean domain. Show that for nonzero α, β ∈ R there are γ, λ, μ ∈ R such that γ | α, γ | β and γ = λα + μβ. (9)

(b) Show that Z[i] = {a + bi : a, b ∈ Z} is a Euclidean domain. Let α = 149 and β = 44 + i. Find a greatest common divisor γ for α and β in Z[i]. Also find λ, μ ∈ Z[i] with γ = λα + μβ. (11)

[20]

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