225 2010

LANCASTER UNIVERSITY 2010 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 the six elements u, v, w, x, y and z. ∗ u v w x y z u ? v w v x y u z z ? Find the identity of G, and determine which elements must appear in the positions indicated by question marks. [Hint: apply associativity to the expression wyx; it may help to consider inverses.] [6] A2. Let π = ( 1 2 3 4 5 6 7 8 9 6 1 8 7 9 2 5 3 4 ) and σ = (1 4 3 9 6 2)(5 8 7)(2 3 8 7)(4 9 5 6). (a) Write π and σ in cycle notation. [2] (b) Write π and σ as products of transpositions. [2] (c) Calculate sign(π) and sign(σ). [2] (d) Determine the orders of π and σ. [2] please turn over 1 SECTION A continued A3. Let G be a group with identity e. Define a subset of G by H = {x ∈ G : x2 = e}. (a) If G is abelian show that H is a subgroup of G. [3] (b) Determine H when G is the group S3. Is H a subgroup in this case? [3] A4. For each of following ten statements, decide whether it is true or false; no justification of your answers is required. [10] (a) A group of order 43 can have an element of order 7. (b) Every cyclic group is abelian. (c) Every abelian group is cyclic. (d) If H is a subgroup of a group G and x, y ∈ H then we must have xy−1 ∈ H. (e) The kernel of a group isomorphism must be trivial. (f) 2̂X − 1 = X̂2 − 1 in the quotient ring Z[X]/I, where I = {a1X + a2X2 + · · · + anXn : n ∈ N, a1, a2, . . . , an ∈ Z}. (g) S ∼= R/ kerϕ whenever R and S are rings and ϕ : R → S is a ring homomorphism. (h) Z91 is a field. (i) 1̂3 is the multiplicative inverse of the element 3̂ in the field Z19. (j) The ring Z9 is an integral domain. please turn over 2 SECTION A continued A5. Let S = ⎧⎪⎪⎨ ⎪⎪⎩ ⎛ ⎜⎜⎝ a 0 0 b 0 0 c d e ⎞ ⎟⎟⎠ : a, b, c, d, e ∈ R ⎫⎪⎪⎬ ⎪⎪⎭ . (a) Show that S is a subring of M3(R). [4] (b) Show that the map ϕ : S → S given by ϕ ⎛ ⎜⎜⎝ a 0 0 b 0 0 c d e ⎞ ⎟⎟⎠ = ⎛ ⎜⎜⎝ e 0 0 0 0 0 0 0 a ⎞ ⎟⎟⎠ is a ring homomorphism, whereas the map ψ : S → S given by ψ ⎛ ⎜⎜⎝ a 0 0 b 0 0 c d e ⎞ ⎟⎟⎠ = ⎛ ⎜⎜⎝ a 0 0 b 0 0 0 d e ⎞ ⎟⎟⎠ is not a ring homomorphism. [7] (c) Find the kernel and the image of the ring homomorphism ϕ defined in (b). [3] A6. Find the multiplicative inverse of the element 2̂1 in the field Z29, and use it to find all elements x ∈ Z29 which satisfy 2̂1x = 6̂. [6] please turn over 3 SECTION B B1. (a) Let S = R \ {−1}. Show that a ∗ b = ab + a + b defines a binary operation on S, and that under this binary operation S is an abelian group. [8] (b) Let G be a group with identity e. If a, b ∈ G satisfy ab = e show that b is the inverse of a. Using this or otherwise, show that (xy)−1 = y−1x−1 for all x, y ∈ G. [4] (c) Consider the following Cayley table of a group G of order 8. ∗ e a b c d f g h e e a b c d f g h a a b c e h d f g b b c e a g h d f c c e a b f g h d d d f g h e a b c f f g h d c e a b g g h d f b c e a h h d f g a b c e (i) Find x, y ∈ G which solve the equations a.x = b and f.y.g = h. [2] (ii) Determine the orders of the elements b, c and g. [3] (iii) State Cayley’s Theorem. Find the permutations associated with the elements a and d by Cayley’s Theorem. [3] please turn over 4 SECTION B continued B2. (a) Let G be a finite group, and H a subgroup of G. (i) Define what is meant by a right coset of H in G. [1] (ii) The relation R is defined on the elements of G by aR b ⇐⇒ ab−1 ∈ H. Show that R is a equivalence relation, and the equivalence classes are the right cosets of H. [5] (iii) Prove that |H| divides |G|. [3] (b) Let G and H be groups, and ϕ : G → H be a group homomorphism. (i) Define the kernel and image of ϕ. [2] (ii) Let h ∈ imϕ. Prove that the set {g ∈ G : ϕ(g) = h} is a right coset of kerϕ. [3] (c) A group homomorphism ϕ : Z12 → Z12 is given by ϕ(n̂) = 9̂n for all n̂ ∈ Z12. (You do not need to justify that ϕ is a homomorphism.) (i) Identify the kernel K and the image I of ϕ. [2] (ii) For each h ∈ I, determine {g ∈ Z12 : ϕ(g) = h} and identify this set as a right coset of K. [4] please turn over 5 SECTION B continued B3. Let e = {1, 2, 3}, and let R be the set of all subsets of e, that is, R = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, e}. For a, b ∈ R, we define a+ b = (a ∪ b) \ (a ∩ b) (so that a+ b is the set of all elements which belong to either a or b, but not to both) and a× b = a ∩ b. (a) Calculate {1, 2} + {2, 3} and {1, 2} × {2, 3}. [3] (b) Explain why addition in R is commutative. [2] (c) Show that the empty set ∅ is a zero element in R. [2] (d) Show that every element a ∈ R satisfies a + a = ∅, and deduce from this that a has an additive inverse. [3] In the remainder of this question, you may suppose without any further proof that R is a commutative ring with respect to these operations. (e) Does the ring R have a multiplicative identity? Justify your answer. [3] (f) Prove that I = {∅, {1}, {2}, {1, 2}} is an ideal in R. [7] B4. (a) Define Q[i] = {p + qi : p, q ∈ Q}. (i) Prove that Q[i] is a subfield of C. [5] (ii) Does the ring Q contain a subring isomorphic to Q[i]? Justify your answer. [5] (b) Let F be a field. (i) Show that F is simple. [5] (ii) Let R be a ring, and let ϕ : F → R be a non-zero ring homomorphism. Show that ϕ is injective. [Hint: use (i).] [5] end of exam 6