Let G = (Z/6Z, +) and H = C12 = = {e, a, . . ....

Question

Question

Let G = (Z/6Z, +) and H = C12 = = {e, a, . . . , a^11}. Define a homomorphism φ : G → H by φ([1]6) = a^4.

a. Determine K, the kernel of φ, as a subgroup of G (Hint: you will want to compute φ([j]6) for all the elements [j]6 ∈ G.)

b. Determine the image of φ as a subgroup of H.

c. Determine the factor group G/K. By this I mean: write down the elements of the group G/K and its multiplication table.

Answer 1

Answer #1

ct φ(1) = a4 6(2) = φ(1+1) = φ(1) φ(1)-๕๕ = a8 φ(3) = e φ(4)-44 φ(5) = a 6(6) = e K-(0,3)

Answer 2

Similar Homework Help Questions

quention for 8 iz) 23)1Dy ave 7. (10M) Prove that o: Z x Z Z given...

quention for 8 iz) 23)1Dy ave 7. (10M) Prove that o: Z x Z Z given by (a, b) a+b homomorphism and find its kernel. Describe the set is a 8. (10M) Prove that there is no homomorphism from Zs x Z2 onto Z4 x Z 9.(10M) Let G be a order of the element gH in G/H must divide the order of g in G. finite group and let H be a normal subgroup of G. Prove that (16M)...
Problem 3. Subgroups of quotient groups. Let G be a group and let H<G be a...

Problem 3. Subgroups of quotient groups. Let G be a group and let H<G be a normal subgroup. Let K be a subgroup of G that contains H. (1) Show that there is a well-defined injective homomorphism i: K/ H G /H given by i(kH) = kH. By abuse of notation, we regard K/H as being the subgroup Imi < G/H consisting of all cosets of the form KH with k EK. (2) Show that every subgroup of G/H is...
Answer Question 5 . Name: 1. Prove that if N is a subgroup of index 2...

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...

Let a : G + H be a homomorphism. Which of the following statements must necessarily...

Let a : G + H be a homomorphism. Which of the following statements must necessarily be true? Check ALL answers that are necessarily true. There may be more than one correct answer. A. If kera is trivial (i.e., ker a = {eg}), then a is injective. B. If the image of a equals H, then a is injective. C. The first isomorphism theorem gives an isomorphism between the image of a and a certain quotient group. D. The first...
Only need answer from (IV) to (VI) Only need answer from (IV) to (VI) Math 3140...

Only need answer from (IV) to (VI) Only need answer from (IV) to (VI) Math 3140 page 1 of 7 1. (30) Let R be the group of real numbers under addition, and let U = {e® : 0 E R} be the group of all complex numbers on the unit circle under multiplication. Let o: R U be the map given by = e is a homomorphism of groups. (i) Prove that (i) Find the kernel of . (Don't...
ei0 : 0 E R} be the group of all complex numbers on the unit circle...

ei0 : 0 E R} be the group of all complex numbers on the unit circle under multiplication. Let ø : R -> U 1. (30) Let R be the group of real numbers under addition, and let U be the map given by e2Tir (r) (i) Prove that d is a homomorphism of groups (ii) Find the kernel of ø. (Don't just write down the definition. You need to describe explicit subset of R.) an real number r for...

Question 4 Exercise 1. Let G be a group such that |G| is even. Show that...

Question 4 Exercise 1. Let G be a group such that |G| is even. Show that there exists an EG,17e with x = e. Exercise 2. Let G be a group and H a subgroup of G. Define a set K by K = {z € G war- € H for all a € H}. Show that (i) K <G (ii) H <K Exercise 3. Let S be the set R\ {0,1}. Define functions from S to S by e(z)...
4. Let G be a group. An isomorphism : G G is called an automorphism of G. (a) Prove that the set,...

the following questions are relative,please solve them, thanks! 4. Let G be a group. An isomorphism : G G is called an automorphism of G. (a) Prove that the set, Aut(G), of all automorphisms of G forms a group under composition. (b) Let g E G. Show that the map ф9: G-+ G given by c%(z)-gZg", įs an automorphism. These are called the inner automorphisms of G (c) Show that the set of all g E G such that Og-Pe...
1. Let G - Z. Let H - {0,3,5,9) be a subgroup of (you do not...

1. Let G - Z. Let H - {0,3,5,9) be a subgroup of (you do not need to prove this is a subgroup of G). Prove that G/l is a valid quotient group. Explain what the elements of G/H are and what the group operation is. 2. Let G be a group and H a normal subgroup in G. I E H for all IEG, then prove that G/H is abelian