(a) Express the permutation (2 4 5)(1 3 5 4)(1 2 5) as a single cycle or as a product of cycles. (b) How many elements of the permutation group Se map 2 to 2 and 5 to 5, while the re- maining numbers in the set S= {1,2,3,4,5,6} are free to permute? In the Cartesian plane = {(x,y): x,y € R} consisting of points with rectangular coordinates (x, y), define the relation by (x1, y1)~ (x2, 92) 21x₂. (c) Prove that is an equivalence relation on the set P. (d) Describe the equivalence classes geometrically.

Q: Let (x, y) stands for a complex number z E C where x = Re(z) and y = Im(z). Now, let us define a new…

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Q: 14. Let A = {1,2,3,4,5,6} and p₁ = (4,2,5) and p₂ = (2,6,1) be permutations of A. Compute p₂¹P₁-1…

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Q: 5-) Find all elements in his which has multiplcn the interse, then find their inverses

A: The objective of the question is determine the elements of Z15  and evaluate the   multiplicative…

Q: Q2(a) Consider the Cartesian product S = A x B = {0, 1, 2} × {0, 1, 2, 3, 4} as a raster of…

A: Given that S= {1,2,3}×{0,1,2,3,4} We need to find the subset of S.

Q: Describe geometrically (line, plane, or all of R 3) all linear combinations of (a) [ 1/2/3 ] and […

A: We know that a set of vectors u1, u2, ……, un is said to be linearly independent if the equation…

Q: Write the permutation corresponding to R90 in the left regular representationof D4 in cycle form.

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Q: The permutations p = (3 5 4), q= (1 5 2 3) and r = (1 5 2) are elements of S5. pal, m. (a) Find each…

A: (a) To find each of the given permutations in cycle form, we will apply the permutations…

Q: 1. Find an n-tuple representation for the coordinates of [1 3] with respect to the sets given (b)…

A: We have to find an n-tuple representation for the coordinates of 13 w.r.t the sets given: a 10, 11

Q: The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13.. are defined by the recursion formula xn+1 = Xn + xn-1,…

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Q: Consider the set C4 = {i, -1, -i, 1}, the fourth roots of unity in the complex numbers. (a) Make a…

A: The given set is C4=i,-1,-i,1, the fourth root of unity in the complex numbers. We know that Cayley…

Q: 2 3 4 5 1 3 5 (1 2 3 4 5 Consider the permutation 12 4 2. 3 4 1 1. Express as a product of disjoint…

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Q: Let f= (14723 5 6) and g = (1 5 3 264) be permutations in S. What is the second line of fin two-line…

A: Introduction: Permutation businesses are mathematical businesses whose individuals are permutations…

Q: 2) Let A = {1,2,...,n} and let Sn denote the permutations on A. A 2-cycle of the form (a b) is…

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Q: Fifteen balls, including three each of five different colors, are arrange in a triangle as shown.…

A: “Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts…

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

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Q: 4 Let f = (4 5 6) and g = (1 9 8 4) (2 7 5) (3 6) be two permutations in S9. (a) Compute fogo f-¹,…

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Q: 2. Verify and show whether each of the following is a groupoid, semigroup, monoid, or N/A in case…

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Q: 4. (i) Write the following permutations in S9 as a product of disjoint cycles 6 1 2 3 4 5 6 7 8 9 7…

A: Note: As per our guidelines, we will solve first three sub-parts. Please repost remaining parts.

Q: Activity 25.22. One advantage of decomposing a permutation as a product of disjoint cycles is that…

A: The idea of permutation will be used in solving this problem

Q: In general, if (12 · · · k) is a cycle of length k and o is any permutation then o(12 ·.· k)o -1 =…

A: Given: T=(12.....k)

Q: How many ways are there to put five beads on a necklace if there are eight distinct beads to choose…

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Q: - Let 1 2 3 4 5 6 7 8 2 3 5 6 7 1 8 4 1 2 3 4 5 6 7 8 4 8 3 6 1 2 5 7 and be permutations on the set…

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Q: 2 345 6 7 1/ 168 LEt a= 2. 1. 3457 a)Write a as a product of disjoint cycles. b) write a as a…

A: First we have to write alpha as a product of disjoint cycles-

Q: 10. Construct the operation table for each of the following: (a) (Zg, +), (Zg, ), and (Ug, ) 11

A: Introduction: “Since you have posted multiple questions, we will provide the solutiononly to the…

Q: [1 2 3 4 5 5 2 3 4 1 [1 2 3 4 5] 1 5 4 3 2 1. Let o = and T = be permutations in S6. (a) (2 points)…

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Q: What are all of the distinct generators of Z17 with respect to multiplication? What are all the…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: Which of the following transformations results in a congruent preimage and image for a figure?

A: Given query is to find which transformation given congruent preimage and image for a figure.

Q: List all elements of U(10) and give a multiplication table for the group U(10)

A: Given  that, The group U(10). We have to find the all elements in U(10) and multiplication table for…

Q: entire elements of D8 in terms of products of disjoint cycles in S8 ?

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Q: Show that for A and B of n×n order with complex enteries the matric Ab need not be similar to BA ,…

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