Download Six Elements - Groups and Rings - Exam and more Exams Statistics in PDF only on Docsity!
LANCASTER UNIVERSITY
2009 EXAMINATIONS
PART II (Second Year) MATHEMATICS & STATISTICS 2 hours Math 225: Groups and Rings
You should answer ALL Section A questions and THREE Section B questions. In Section A there are questions worth a total of 50 marks, but the maximum mark that you can gain there is capped at 40. SECTION A
A1. The following is an incomplete Cayley table for a group G with six elements u, v, w, x, y and z. โ u v w x y z u v? w z u x? y v x z Find the identity of G, and determine which elements must appear in the positions indicated by question marks. [Hint: it may help to consider inverses.] [6]
A2. Let G be the multiplicative group of invertible 2 ร 2 matrices with real entries. Define
H =
{(a b 0 c
: a, b, c โ R, ac = 1
and K =
{( 1 a b 1
: a, b โ R, ab = 0
(i) Show that H is a subgroup of G. [4] (ii) Show that K is not a subgroup of G. [2]
please turn over
SECTION A continued
A3. Let ฯ = (^ 1 2 3 4 5 6 7 8 99 1 7 5 3 2 4 6 8^ )^ and ฯ = (1 2)(3 4 5)(6 7 8 9)(1 2 3 4 5)(6 7 8 9 1 2).
(a) Write ฯ and ฯ in cycle notation. [2] (b) Write ฯ and ฯ as products of transpositions. [2] (c) Calculate sign(ฯ) and sign(ฯ). [2] (d) Define the alternating group A 9 ; without calculating the product ฯฯ, show that ฯฯ โ A 9. [2]
A4. For each of the following ten statements, decide whether it is true or false; no justification of your answers is required. [10] (a) For each positive integer n, there is at least one abelian group of order n. (b) The intersection of two subgroups of a group G is always a subgroup of G. (c) In every cyclic group every non-identity element is a generator. (d) A group of order 42 can have a subgroup of order 4. (e) Any two groups of order 6 are isomorphic. (f) The rings Z 2 and Z 3 are isomorphic. (g) The subrings 2Z and 3Z of Z are isomorphic. (h) The product of two cubic polynomials over Z has degree 9. (i) The polynomial ring Z 2 [X] is an integral domain. (j) There is an integral domain of characteristic 39.
please turn over
SECTION B
B1. (^) (a) Let S = {(a, b) โ R^2 : a = 0}. Show that (a, b) โ (c, d) = (3ac, 3 ad + b) defines a binary operation on S, and that under this binary operation S is a group which is not abelian. [9] (b) Let G be a finite group. (i) Let g โ G. State how to determine the order of g by examining its powers. [1] (ii) Show that, for all g, h โ G and n โ N, we have (hghโ^1 )n^ = hgnhโ^1. Deduce that o(hghโ^1 ) = o(g). [4] (iii) Deduce from (ii) that, for all a, b โ G, the elements ab and ba have the same order. [2] (c) Determine the order of the permutation ฯ = (1 4 3 5)(2 8)(2 8 3 4 5 9 7)(5 6 7) and list its distinct powers. [4]
please turn over
SECTION B continued
B2. (^) (a) Let G be a group, and H a subgroup of G. (i) Define what it means for H to be normal. [1] (ii) Suppose that H is normal, and let G/H denote the set of right cosets of H. Show that Hx.Hy = Hxy for all Hx, Hy โ G/H defines a binary operation on G/H. [3] (b) Let G and H be finite groups, and ฯ : G โ H a group homomorphism. (i) Define the kernel and image of ฯ. [2] (ii) State without proof the Fundamental Isomorphism Theorem for groups as it applies to ฯ. [1] (iii) Show that |G| = |ker ฯ| |im ฯ|. [2] (c) Show that any group homomorphism ฯ : Z 42 โ Z 25 must be trivial. [3] (d) The map ฯ : Z 15 โ Z 6 is defined by ฯ(ฬn) = 4 ฬn for all ฬn โ Z 15. (i) Show that the map ฯ is well-defined, and is a group homomorphism. [2] (ii) Identify the kernel K and the image I of ฯ. [2] (iii) List the elements in each right coset of K. [2] (iv) Give the Cayley tables of the group I and the quotient group Z 15 /K. [2]
please turn over
