2 (2+2+1 marks) Consider the function GL(2,R-R A det A a) Prove that f is a surjective homomorphism. b) Verify that N-AL()dAE Ois a nomal subgroup of GL(2.R) GL(2.Ra group? a group? If so, with w...
1- (2,5+2,5 mark) Consider in GL(2, Q), the subset (a=1 or a=-1),bez Prove that H, with multiplication, is a subgroup of GL(2,Q) a) Is the function b) an homomorphism of groups? Justify your answer 2 (3 marks) Let G be a group and a E G an element of order 12. Find the orders of each of the elements of (a) 3- (1+1,5 marks) Let G be a group such that any non-identity element has order 2. Prove that a)...
3 -(3+2 marks) Consider, with the usual operations, the groups ZxZ and Zxz, a) Prove that a-b.lb (a,b) → (a-시4) is a surjective homomorphism b) Show that ZxZ (z.xz)-zxz, (22)Zxz, 3 -(3+2 marks) Consider, with the usual operations, the groups ZxZ and Zxz, a) Prove that a-b.lb (a,b) → (a-시4) is a surjective homomorphism b) Show that ZxZ (z.xz)-zxz, (22)Zxz,
10. Let G = D. be the dihedral group on the octagon and let N = (r) be the subgroup of G generated by r4. (a) Prove that N is a normal subgroup of G. (b) If G =D3/N, find G. (c) Using the bar notation for cosets, show that G = {e, F, 2, 3, 5, 87, 82, 83}. Hint: Show that the RHS consists of distinct elements and then use part (b). (d) Prove that G-D4. Hint: It...
Answer Question 5 . Name: 1. Prove that if N is a subgroup of index 2 in a group G, then N is normal in G 2. Let N < SI consists of all those permutations ơ such that o(4)-4. Is N nonnal in sa? 3. Let G be a finite group and H a subgroup of G of order . If H is the only subgroup of G of order n, then is normal in G 4. Let G...
Exercise 41.2 Consider the signal f(t) window w(t)e amat, α E R, and the Gaussian (a) Verify that is well-defined (even though f f L'(R)). (b) Compute Ws(A, b) using the following result: e-ra(1+iz)2 dt = a-2 For a > 0 and E R, (c) Show that l W, (A, b)12 attains its maximum when λ a. Exercise 41.2 Consider the signal f(t) window w(t)e amat, α E R, and the Gaussian (a) Verify that is well-defined (even though f...
Question 8 (6 marks) Consider the function f: [-2, 2] shown in the diagram. R whose graph is (a) State the largest intervals on which f is injective f(z) (b) For each interval I you found in (a), sketch the graph of the inverse g of the restriction of f to I and state the domain and range of g (c) If we know that f(-x) = f(x) for all r e dom(f), what can we conclude about the relationship...
Problem 1. Consider the function f(x)- 3.12 show that f is Riemann integrable on [0.2] and use the definition to find .后f(x)dr Problem 2. Consider the function -2, zEQ 2, O f(r) = Show that f is not Riemann integrable on 0,1 but s Reemann integrable on this interval. Problem 3. (a) Let f be a real-valued function on a, b] such thatf()0 for all c, where c E [a, b Prove that f is Riemann integrable on a, b...
5. Consider the function f(z)-1. (a) Sketch the horizontal line y 1/2 together with its image under f b) Verify that the image of line y- b>0 is a circle. What are its center and radius? c) What is the image of the half-plane : y>1/2 under f? 5. Consider the function f(z)-1. (a) Sketch the horizontal line y 1/2 together with its image under f b) Verify that the image of line y- b>0 is a circle. What are...
Please detail all your answers 2. Consider the function f : {1, 2, 3, 4, 5} → {1, 2, 3, 4} given by the table below: (15 points) x 1 2 3 4 5 f (x) 3 2 4 1 2 (a) Is f injective? Explain. (b) Is f surjective? Explain. (c) Write the function using two-line notation. 5. In the game of Hearts, four players are each dealt 13 cards from a deck of 52. Is this a function?...