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 ma...
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 what operation? c) Is 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 what operation? c) Is
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)...
2. Let p: G -G be a surjective group homomorphism (a) Show that if G is abelian then G' is abelian. (b) Show that if G' is cyclic then there is a surjective homomorphism from (Z, +, 0) to G'. (Hint: use the fact that Z is generated by 1 and G' has a generator). (c) Use Part (a) and (b) to show that every cyclic group is abelian.
b=2019a^3+2663a a=? I'm trying to prove the equation is surjective
(4) Let p Z be a prime. Prove that zli/(p+1) has exactly ] p2 +1 elements. Use that 5+5i (2+i)(3+i) to determine how many elements Zu/5+5i) has. (5) Let m,n be integers with m|n. Prove that the surjective ring homomor- phism Z/n -> Z/m induces a group homomorphism on units, and that this group homomorphism is also surjective. (4) Let p Z be a prime. Prove that zli/(p+1) has exactly ] p2 +1 elements. Use that 5+5i (2+i)(3+i) to determine...
Thanks 3. Suppose that S:G - H is a homomorphism of groups and that S is a normal subgroup of H. Then show that {r EG (2) ES) is a normal subgroup of NB: First show that {x EG f(x) E S} is a subgroups of G
Surjection. Prove F: A => 13 is surjective . YE B. Z = A - F (17.3) CA 1. Show : 9:Z=> 13 - {Yo], given g(x)= fx) for X6 Zis well-defined function 2. Shour: g is surjective
3. Let y: K + Aut(H) be a homomorphism. Write (k) = Ok. Let G be a group. A function d: K + H is called a derivation if dikk') = d(k) (d(k')). Show that d: K + H is a derivation if and only if V: K + H y K given by v(k) = (d(k), k) is a homomorphism. 4. Suppose that a: G + K is a surjective homomorphism and that 0: K + G is a...
Question 3 (Chapter 6) 13+2+3+6 14 marks Fix p EN and consider the following set: : T1 (a) Prove that Cp is convex. (b) Prove that C, is a cone. (c) Compute Ci and C2. (d) Show that x = 0 is an extreme point of CP. Question 3 (Chapter 6) 13+2+3+6 14 marks Fix p EN and consider the following set: : T1 (a) Prove that Cp is convex. (b) Prove that C, is a cone. (c) Compute Ci...
(1) Consider Z with the addition and multiplication mod 3 as usual. Let R=ZgxZg. Define (a, b)+ (a',b) (aab+and(a,b)((aa-bab +a'b) (a) Show that (R, +) is a commutative ring. b) Show that (1,0) is the identity element for the multiplication. c) Show that the equation 22 hs exactly two solutions in R Bonus Problem) Show that (R, +,.) is a field. (Hint: To find multiplicative inverse, first show that a2 + b2メ0 if (a, b)メ(0.0). Then compute (a, b).(a,-b).) (1)...