(c) In this part of the question we consider U25 as a group under multiplication. Let a E Z. Suppose that 5 a. (1) Suppose that [al0]25 [1]25 and [a*]25 # [1]25. Prove that (la)25) = U25. %3D (ii) Show that ((2]25) = U25. (ii) Suppose that ([a]25) = U25. Show that ([a']25) = U25. %3D %3D

Q: 11. Let K be the splitting field of x³ – 2 over Q. (Refer to Example 50.9.) a. Describe the six…

A: please see the next step for solution

Q: Let (a1a2a3a4) and (a5a6) be disjoint cycles in S10. Show that there isno element x in S10 such that…

A: Given: a1a2a3a4 and a5a6 are disjoint cycles in S10.We need to show that there is no element x in…

Q: 7. Consider an element z = the order of r is the least common multiple of the orders of its…

A:

Q: 9. Describe the group of the polynomial (x* – 1) e Q[x] over Q.

A:

Q: 12. Describe the group of the polynomial (x* – 5x? + 6) € Q[x] o over Q.

A: please see the next step for solution

Q: 17. Compute d = ged(28, 124). Find integers x, y so that 28x + 124y = d. 18. Find the order of the…

A:

Q: 7. [10] Let G be a group, with x € G and o (x) = 48. Give the order of the following elements of G:…

A:

Q: Which one of the following is not a group? (1): (8Z,+) where 8Z = {8n | n € Z} (2): (Z8, +) where…

A:

Q: 1. 3 in U(11), the group of units modulo 11.

A: As per the guidelines we can answer the first question since u have posted multiple questions.

Q: [S,*]= [Z,+] and [S',*']=[Z,∘] where a∘b=a+b-1. Then ?(n)=n+1 is an isomorphism from [Z,+] to [Z,∘]

A:

Q: (a) Make addition and multiplication tables for Z/3Z.

A: “Since you have posted multiple questions, we will provide the solution only to the first question…

Q: (a) Explain why any automorphism (b) Show that any automorphism that Aut (S3) 6. of S3 takes a…

A:

Q: (a) (Z,*) with a × b = a/b is (b) (Q, *) with a * b = ab - a - b is (c) (Q,*) with a*b = a² +6² is…

A:

Q: A All B (ii), (iii) D E None (i), (iii) (i), (ii) Suppose such that (i): (ii): : Z→ Z40 is a group…

A: Please find attachment

Q: (3) Find the centralizer c(A) in the group GL₂(IR). A [1

A: Let  be a fixed element of a group . The centralizer of  in , denoted by , is the set of all…

Q: Show that there are no simple groups of order 255 = (3)(5)(17).

A:

Q: 8. Let S = (12345), T = (13)(45) in S5. Describe the group generated by S and T?

A: Sol:- The group generated by S and T is the group {S, T} = ⟨S, T⟩. We first observe that S and T…

Q: (a) Let a and b be two elements of Sn with the same cycle structure. Explain how to produce an…

A: Good question. Have answered all the details in step 2.

Q: 2.10 Let G be the set of all 2×2 matrices ( .), where a, b ER and a² + b² +0. Show that G forms a…

A:

Q: If the given three vectors are coplanar, then find the value of x. A = i-2j + 3k, B = xj + 3k, C =…

A: A, B, C are coplanar.So

Q: 3. Prove that G = {a+b√2: a, b € Q and a and b are not both zero} is a subgroup of R* under the…

A: Result: Let G, * be a group and H is a subset. The subset H is said to be subgroup, if for every a,…

Q: We multiply α (alpha) by 2 if we have 2 groups True False

A: We multiply α (alpha) by 2 if we have 2 groups False. Alpha (α) is the significance level of a…

Q: * 3. In each case, determine if the statement is true or false. (a) For all [a] e Z6, if la] [0],…

A: