Please solve question 3 and question 4 12:37 A Mf Math365 Assign... /Exercise 4.Consider the following permutations in S9(12 3 4 5 6 7 8 9(1 2 3 4 5 6 7 8 9and r=(35 2 1 4 6 9 785 2 6 7 1 3 8 91) Write o,r, oand1' in cycle notation.2) Write o andas a product of 2-cycles.3) What is | o |? What is | 7|? And what is | ot |?4) Compute or .5) Find oro-, Write your answer in cycle notation.6) Find a by solving oa =r. Write your answer in cycle notation.2/2II... Exercise 2.What cycle is (a,a2 ... an)¯1?Exercise 3.Part A: Let G = Sn and let H = {a € G|a(1) = 1}. Show that H is a subgroup of G.Part B: Let S9 be the symmetric group and let a be an element of S9 defined by:1 2 3 4 5 6 7 8 95 4 3 29 8 67 6a =1) Write a as a product of disjoint cycles.2) Determine the order of a.3) Calculate a²2018.4) Find the smallest integer n such that a™+3 = a.5) Prove that a is an odd permutation.6) Let ß = (7 3)(1 5 4). Determine aß-'a.7) Show that for any permutations a and ß in Sn, a*ß and ß have the same parity.

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Description: Please solve question 3 and question 4 12:37 A Mf Math365 Assign... /Exercise 4.Consider the following permutations in S9(12 3 4 5 6 7 8 9(1 2 3 4 5 6 7 8 9and r=(35 2 1 4 6 9 785 2 6 7 1 3 8 91) Write o,r, oand1' in cycle notation.2) Write o andas a

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Exercise 4. Consider the following permutations in S, _(1 2 3 4 5 6 7 8 9) (3 5 2 1 4 6 9 7 8) (1 2 3 4 5 6 7 8 9 and r= 5 2 6 7 1 3 89 4 1) Write o, r, a" and r' in cycle notation. 2) Write a and r as a product of 2-cycles. 3) What is |a|? What is |r|? And what is | or |?

Exercise 4. Consider the following permutations in S, _(1 2 3 4 5 6 7 8 9) (3 5 2 1 4 6 9 7 8) (1 2 3 4 5 6 7 8 9 and r= 5 2 6 7 1 3 89 4 1) Write o, r, a" and r' in cycle notation. 2) Write a and r as a product of 2-cycles. 3) What is |a|? What is |r|? And what is | or |?

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.6: Quotient Groups

Problem 3E: In Exercises , is a normal subgroup of the group . Find the order of the quotient group . Write out...

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