MAS2006

UNIVERSITY OF EXETER SCHOOL OF MATHEMATICAL SCIENCES

GROUPS, RINGS AND FIELDS

June 2001  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. Approved calculators of the following type may be used Casio fx82 series, Sharp EL521 or EL531 Series and Texas TI-30X or TI-36X.

SECTION A

1. (a) Draw a regular hexagon (six sides). Describe each of its symmetry operations, and give the order of each. Find all four-element subgroups of the symmetry group of the hexagon. (7)

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

(i) (1 8 3)(2 4 7 5 6); (ii) (1 12 7)(2 11 6 5 9 10 3)(4 8); (iii) (1 13 7)(2 15 6 11 9 10)(3 14)(4 8 5 12).

(6)

(c) Let G be a group with operation written multiplicatively. Show that (ab)^{−1} = b^{−1}a^{−1} for each a, b ∈ G. (3)

(d) Show that the set of units in a ring is a group with the operation of multiplication. Find the units in the ring Z_{20} of integers modulo 20. For each unit find its inverse and its order. (8)

(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 & 0 \\ c & d \end{matrix} ) : a, c, d ∈ R\};

(ii)

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

(7)

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

(i) a = 61, b = 44; (ii) a = 618, b = 429;

(iii) a = 1053, b = 403. (9)

[40]

SECTION B

2. (a) Give the definitions of the terms group and subgroup. (4)

(b) Which of the following sets G are groups with respect to the operation ∗? Justify your answers.

(i) G = Z, a ∗ b = a − b; (ii) G = R, a ∗ b = a + b + 2;

(iii) G = {a ∈ R : a ≠ 0}, a ∗ b = −2ab. (8)

(c) 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^{∗} : Re(z) > 0} (where Re(z) denotes the real part of the complex number z);

(ii) G = Z with the operation of addition,

H = {a ∈ Z : a is even}; (iii) G = Q^{∗} with the operation of multiplication,

H = {m/n : m, n both odd integers}.

(8) [20]

3. (a) State and prove Lagrange’s theorem. (10)

(b) Find the left and right cosets of the following subgroup of S_{4}: H = {id, (1 3 2 4), (1 2)(3 4), (1 4 2 3)}. (6)

(c) Suppose that H is a subgroup of G and that the operation in G is written as multiplication. If aha^{−1} ∈ H for all a ∈ G and h ∈ H, show that each left coset of H in G is a right coset of H in G. (4)

[20] 4. (a) Let G and H be groups. What is a homomorphism from G to

H? Let ϕ : G → H be a homomorphism of groups. Suppose that the operations in G and H are written multiplicatively and that the identities of G and H are denoted by e_{1} and e_{2} respectively. Show that ϕ(e_{1}) = e_{2} and that ϕ(a^{−1}) = ϕ(a)^{−1} for each a ∈ G. (8)

(b) What is the image of a homomorphism? Show that the image of a homomorphism from G to H is a subgroup of H. (4)

(c) In each of the following cases determine whether ϕ : G → H is a homomorphism. Justify your answers.

(i) G = R with the operation of addition, H = C^{∗} with the operation of multiplication, ϕ(x) = cos x − i sin x;

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

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

(8) [20]

5. (a) What is an integral domain? What is a Euclidean function on an integral domain? Show that v(a+b\sqrt{−2}) = |a+b\sqrt{−2}|^{2} = a^{2}+2b^{2} is a Euclidean function on Z[\sqrt{−2}] = \{a + b\sqrt{−2} : a, b ∈ Z\}. (10)

(b) What is an irreducible element in an integral domain? Show that 3, 2 + \sqrt{−5} and 2 − \sqrt{−5} are all irreducible elements of R = Z[\sqrt{−5}] = \{a + b\sqrt{−5} : a, b ∈ Z\} and deduce that

9 = 3 × 3 = (2 + \sqrt{−5})(2 − \sqrt{−5})

are two essentially distinct factorizations of 9 into irreducible elements in R. (10)

[20]

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