Exercise 3.Part A: Let GS, and let H = {a € Gla(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 S, defined by:α =1 2 3 4 5 6 7 8 9 5 4 3 2 9 8 6761) Write a as a product of disjoint cycles.2) Determine the order of a.3) Calculate 20184) Find the smallest integer n such that q”+³5) Prove that a is an odd permutation.= a.6) Let B = (7 3)(1 5 4). Determine aß-¹a.7) Show that for any permutations a and B in Sn, a B and B have the same parity. Exercise 3.Part A: Let G = Sn and let H= {a € G|a(1) = 1}. Show that H is a subgroup of G.%3DPart B: Let S, be the symmetric group and let a be an element of S9 defined by:1 2 3 4 5 6 7 8 9a = (5 4 3 29 8 6 7 61) Write a as a product of disjoint cycles.2) Determine the order of a.3) Calculate a2018.4) 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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Answered: Exercise 3. Part A: Let G = Sn and let…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Exercise 3. Part A: Let GS, and let H = {a € Gla(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 S, defined by: α = 1 2 3 4 5 6 7 8 9 5 4 3 2 9 8 676 1) Write a as a product of disjoint cycles. 2) Determine the order of a. 3) Calculate 2018 4) Find the smallest integer n such that q”+³ 5) Prove that a is an odd permutation. = a. 6) Let B = (7 3)(1 5 4). Determine aß-¹a. 7) Show that for any permutations a and B in Sn, a B and B have the same parity.

Exercise 3.
Part A: Let G = Sn and let H
= {a € G|a(1) = 1}. Show that H is a subgroup of G.
%3D
Part B: Let S, be the symmetric group and let a be an element of S9 defined by:
1 2 3 4 5 6 7 8 9
a = (
5 4 3 29 8 6 7 6
1) Write a as a product of disjoint cycles.
2) Determine the order of a.
3) Calculate a2018.
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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