Problem 6. I Let D2n = {e,r,r², ... n-1 -¹, s, sr, sr²,...,srn-1} be the dihedral group. Prove that N = {e,r,r², ...,rn-¹} is a normal subgroup of D2n and calculate the quotient group D2n/N.

Q: 1. Let M2(R) = {( ) la, b, c, d eR and ad bc + 0}. You already know that M2(R) forms a group under…

A: We have to prove that H is subgroup of M2(R) :

Q: G be and Let then O [a, b'] - [b, a]o ☺ [ai!, b] = [b, a ]a group a, bE G %3D whele xa a xa

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Q: 6. Apply Burnside's formula to compute the number of orbits for the cyclic group G = {(1,5) o (2, 4,…

A: Solve

Q: (a, b | a group of degree 3. Let G 10.1.2. Let Dg b? = e, ba a3b), and let S3 be the symmetric (b) ×…

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Q: 6. Show that for any two elements x, y of any group G, o(xy) = o(yx). %3D

A: Fact 1:In a group G, if x∈G such that xn = e, then O(x)|n (e→ identity element.)i.e. order of x…

Q: Find the (right) cosets for the following subgroups of the stated groups: (i) The subgroup H =…

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Q: 30) Let n 2 3 and k e Zn. Prove that in Dn. p u = up"-k, %3D

A: Note: There are more than one definition for dihedral group. The question is proved using a common…

Q: Find the one-parameter subgroups of Gl(2, R) corresponding to A and B with A 0 (-; d). B = 1 (8J).…

A: As per the question we are given two elements i.e. the 2×2 matrices A, B of the group Gl(2, ℝ) and…

Q: 1. Consider H= (i) and K= (-1) the subgroups of Q, the quaternion group. 1) Explain why H is normal…

A: Basic theory regarding normal subgroups from algebra will be used

Q: 4. The permutations (e, a, B, v) form a group. If e = (1)(2)(3)(4)(5)(6), a = (1)(2)(35)(46), B=…

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Q: 3. Let Dn (n ≥ 3) be the dihedral group of order 2n. (i) Show that D10 ≈ D5 × Z2 by constructing an…

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Q: Consi der a normal subgroup H of order 4 of the dihedral group Dy= (a, b :a* = b*: (ab)* = 1). Then…

A: We know that every left coset of H is equal to its corresponding right coset of H. Using this…

Q: Let G be the group with presentation (x,y: 1² = 1, r² = y²₁ xy = y ¹). Decide how many elements are…

A: Consider the given group G with presentation, x,y:x4=1, x2=y2, xy=yx-1 it can be observed that it is…

Q: Let G = (Z;, x,) be a group then the order of the subgroup of G generated by 2 is О а. 6 O b. 3 О с.…

A: We have to find order of subgroup of G generated by 2.

Q: Let n E Z be an integer number and let H = (n) = {rn,r E Z} i.e\ H = (n) = {.., -3n, –2n, -n, 0, n,…

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Q: (ii) The group D4 of symmetries of a square is given by D₁ = {I, R, R², R³, H,V, D, D'} where I is…

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Q: 6.9. Let G = Cn be a cyclic group of order n, and let X = {x₁,...,n} be a set containing n elements.…

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Q: 5. If H = 122Z and K = 8Z are subgroups of (Z, +). Then H + K = ... %3D

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Q: O Find Subgroup of order a, 3,4 and 6

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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: 7. D4 denotes the dihedral group of order 8. What are the possible homomorphisms P: S4→D4?

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Q: (2,0,5)(1,3)(7,0,3) and t = (2,0)(5,1)(3,7)(0,3). Compute o'r1, giving product of disjoint cycles.…

A: Given: σ=2,0,51,37,0,3 τ=2,05,13,70,3

Q: If the temperature at which a certain compound melts is a random variable with mean value 115°C and…

A: We have to solve given Questions.

Q: Let H be the subgroup of S4 given by H = {e, (1 2), (1 4), (2 4), (1 2 4), (1 4 2)}. (a) What is the…

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Q: Problem 4. Let G = {e,y, …, y"-1, x, xy, .. , xy"-1} be the dihedral group of order 2n where x2 = yn…

A: Given: Dihedral group of order 2n…

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: Q/ In (Z6 , +6 ) find the cyclic subgroup generated by 1, 2, 5.

A: z6={0,1,2,3,4,5}

Q: (a) S' = {z = a + bi e C|a, b € R, |2| = a² + b² = 1} is a subgroup of C*. (b) SO2(R) = {| cos e -…

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Q: Write the differential equation and then determine whether you have overdamped,critically damped,or…

A: Introduction: The differential equation of motion of an unforced damped harmonic oscillation is…

Q: 4-Let (Z12, +12) be a group and let S={4,6}, find subgroup H generated by S. if exist

A: I have used the definition of subgroup generated by a subset.

Q: In each case find the right and left cosets in G of the subgroups H and K of G. (d) G = Z12; H =…

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Q: 27. Prove or disprove that each of the following groups with addition as defined in Exer- cises 52…

A: Let G = Z2xZ4  i.e G = { (0,0),(0,1),(0,2)(,0,3),(1,0),(1,1),(1,2),(1,3)}  Order of G = 8

Q: elements. Why is Dg not isomorphic to Z/8Z?

A: D8 is not isomorphic to Z/8Z because D8 is not a cyclic group but Z/8Z is cyclic. For Cayley table,…

Q: G TE B D E Find the order of the subgroup generated by a for each of the following groups: (124)…

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Q: Consi der a normal Subgroup Hof order 4 of the dihedral group Dy= (a, b :a* = b²: (ab)* = 1). Then…

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Q: 2*. Let Q/Z be the group described in problem 12 of Worksheet 1.1. Find list the elements of the…

A: To identify the subgroups generated by the given elements in the quotient group Q/Z.

Q: 6.14. Let TT € S8 be the permutation defined by π(1) = 3, TT (2) = 8, TT (5) = 6, TT (6) = 1, (a)…

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Q: 1. Prove or disprove that each of the cises 52 of Section 3.1 is cyclic. a. Z2 × Z3

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Q: 7. Compute the group of units U(Z). Is U(Z) cyclic?

A: Q. Compute the group of units U(Z12) . Is U(Z12)  cyclic? Sol. We first compute the group of units…

Q: 1. Let M2(R) = {C )la,b,c, d eR and ad – bc + 0}. You already know . 아 that M2(R) forms a group…

A: A

Q: 3.2 3.3 3.4 Consider the group (Z8, +(mod8)). a) List the elements of (Z8, +(mod8)). b) Find all the…

A: Consider the group (, +(mod8)).We have to (a) List the elements of (,+(mod8))(b) Find all the…

Q: Consider the group D4 = (a, b) = {e = (1), a, a², a³, b, ab, a²b, a³b} %3D where a = (1 2 3 4) and b…

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Q: 8. Let G = U = {z e C| |z| = 1} be the circle group. Then X = C, the set of complex numbers, is a…

A: Given: G=U=z∈ℂ|z=1 is the circle group. X=ℂ, the set of complex numbers, is a G-set with group…

Q: {Ro,R180, D, V} {Ro,R180, D, D'} {Ro,R90, R180, R270} {Ro,R180; H, V}

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