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MMT-003 ASSIGNMENT BOOKLET M.Sc.(Mathematics with Applications in Computer Science) Algebra (July 01, 2009 – June 30, 2010) School of Sciences Indira Gandhi National Open University Maidan Garhi, New Delhi Dear Student, Please read the section on assignments in the Programme Guide for elective Courses that we sent you after your enrolment. As you may know already from the programme guide, the continuous evaluation component has 30% weightage. This assignment is for the continuous evaluation component of the course. Instructions for Formatting Your Assignments Before attempting the assignment please read the following instructions carefully. 1) On top of the ﬁrst page of your answer sheet, please write the details exactly in the following format: ROLL NO : . . . . . . . . . . . . . . . . . . . . . . . . . . . . NAME : . . . . . . . . . . . . . . . . . . . . . . . . . . . . ADDRESS : . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................ ............................ COURSE CODE : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................ COURSE TITLE : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . STUDY CENTRE : . . . . . . . . . . . . . . . . . . . . . . . . . . . . DATE . . . . . . . . . . . . . . . . . . . . . . . . . . . . PLEASE FOLLOW THE ABOVE FORMAT STRICTLY TO FACILITATE EVALUATION AND TO AVOID DELAY. 2) Use only foolscap size writing paper (but not of very thin variety) for writing your answers. 3) Leave a 4 cm margin on the left, top and bottom of your answer sheet. 4) Your answers should be precise. 5) While solving problems, clearly indicate which part of which question is being solved. 6) This assignment is valid only up to June 30, 2010. If you fail in this assignment or fail to submit it by June 30, 2010, then you need to get the assignment for the next year and submit it as per the instructions given therein. 7) The assignment is to be returned to you after being assessed. So, if you want to appear for the December 2009(resp. June 2010)term examination we advise you to submit the assignment at the latest by October 31, 2009(resp. April 30, 2010) so that you can get timely feedback from the counsellor. 8) You will not be allowed to appear for the term end examination without submitting the assignment. We strongly suggest that you retain a copy of your answer sheets. Wish you good luck. 2 Assignment Course Code: MMT-003 Assignment Code: MMT-003/TMA/2009-10 Maximum Marks: 100 1) Which of the following statements are true and which are false? Give reasons for your answer. a) If a ﬁnite group G acts on a ﬁnite set S, then Gs1 = Gs2 for all s1 , s2 ∈ S. 1 0 0 −1 0 0 b) The matrices 0 −1 0 and 0 0 0 are in the same congruence class. 0 0 0 0 0 0 c) If G is a ﬁnite group, ρ : G −→ GL(V) is a group representation on a ﬁnite dimensional vector space V over C and W is a G-invariant subspace of V with V = W ⊕ W, then W is G-invariant. d) If the coefﬁcients aij of an n × n matrix aij are algebraic numbers, the eigenvalues of the matrix are algebraic numbers. √ √ e) F7 3 = F7 5 where F7 is the ﬁnite ﬁeld with seven elements. (10) Z 2) a) Find the automorphism group of , n > 1, n ∈ N. What is its order? (3) nZ b) Use 2 a) to show that n | φ (an − 1) for any positive integers a, n where φ is the Euler’s phi- function deﬁned by φ (m) = |{x ∈ N|x < m, (x, m) = 1}|. (4) c) Show that no group of order p2 q, where p and q are distinct odd primes, is simple. (3) 3) a) Consider the natural action of GL2 (R) on itself by left multiplication. Under this action, show that the stabiliser of x ∈ GL2 (R) is {I}, where I is the 2 × 2 identity matrix, if and only if det(x) = 0. (5) b) Let σ = (i1 , i2 , . . . , ik ) ∈ Sn . Describe the orbits of {1, 2, . . . , n} under the action of σ , the cyclic group generated by σ . (2) c) Let σ = σ1 σ2 · · · σk be the decomposition of σ ∈ Sn as the product of disjoint cycles σ1 , σ2 , . . . σk . Describe the orbits of {1, 2, . . . , n} under the action of σ . (3) 4) a) i) Find a subgroup of GL2 (R) which is isomorphic to C∗ . (2) ii) Prove that, for every n, GLn (C) is isomorphic to a subgroup of GL2n (R). (3) b) Prove that that rule P, A PAP∗ deﬁnes an operation of SL2 (C) on the space W of all 2 × 2 hermitian matrices. Further, show that the function A, A = det (A + A ) − det(A) − det (A ) is a bilinear form on W. (5) 5) Let H be the group of quarternions {±1, ±i, ±j, ±k} and D4 = xi yj x4 = y2 = 1, yxy−1 = x−1 be the dihedral group with 8 elements. In the following exercises, we will show that H and D4 are non-isomorphic, but they have the same character table. a) Find all the conjugacy classes of H and all the conjugacy classes of D4 . (3) 3 b) Show that, if all the irreducible representations of a ﬁnite group are one dimensional, the group is abelian.(Hint: Consider the number of conjugacy classes.) (2) c) Find the degrees of the irreducible representations of H and D4 . (2) d) Find all the irreducible representations of V4 ∼ (Z/2Z) × (Z/2Z), the Klein 4-group. Note that = they are just the homomorphisms from the Klein 4-group to {±1}. (3) e) Note that {±1} is isomorphic to the Klein 4-group. Use φ : H −→ H H {±1} to ﬁnd all the one dimensional irreducible representations of H. (3) i 0 0 1 0 i f) Deﬁne a map ρ : G −→ GL2 (C) by ρ(i) = , ρ(j) = , ρ(k) = . 0 −i −1 0 i 0 Check that ρ is a representation of H. Find the character of ρ and hence check that ρ is a irreducible representation of H. (2) g) Write down the character table of H. (4) D4 ∼ h) Note that x2 is the centre of D4 and that = V4 . Use this to ﬁnd the one dimensional x2 irreducible representations of D4 . (3) 0 −1 1 0 i) Show that ρ deﬁned by ρ(x) = and ρ(y) = is a representation of D4 . Find 1 0 0 −1 its character and hence check that this representation is irreducible. (2) j) Write down the character table of D4 . (4) k) Why are H and D4 not isomorphic? (2) 6) a) If ρ : G −→ GLn (C) is a representation of a group G, the contragredient representation of ρ t is deﬁned by ρ ∗ : G −→ GLn (C) by ρ(g) = ρ g−1 . Check that ρ ∗ deﬁnes a representation of G. If χ ∗ is the character of ρ ∗ show that χ ∗ (g) = χ(g) for all g ∈ G. When will ρ and ρ ∗ coincide? (5) √ √ b) Show that 2 + 3 satisﬁes a polynomial of degree 4 over Q. (2) √ √ c) What is the degree of 2 + 3 over Q? Justify your answer. (3) 7) a) Let F(α) be a ﬁnite extension F of odd degree(greater than 1). Show that F α 2 = F(α). (2) b) Let F ⊂ K and let α, β ∈ K be algebraic over F of degree m and n, respectively. Show that [F(α, β ) : F] ≤ mn. What can you say about [F(α, β ) : F] if m and n are coprime? (5) √ c) Find [Q( 3 2, ω) : Q] where ω 3 = 1, ω = 1. (3) 8) a) If char(F) = 2, show that a polynomial ax2 + bx + c is irreducible iff b2 − 4ac ∈ F∗ 2 where F∗ 2 is the group of squares in F∗ . (2) b) By looking at the factorisation of x9 − x ∈ F3 [x] guess the number of irreducible polynomials of degree 2 over F3 . Find all the irreducible polynomials of degree 2 over F3 . (6) c) If F is a ﬁnite ﬁeld show that there is always an irreducible polynomial of the form x3 − x + a where a ∈ F.(Hint: Show that x → x3 − x is not a surjective map.) (2) 4