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 |?

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Please solve question 3 and question 4
12:37 A M
f Math365 Assign... /
Exercise 4.
Consider the following permutations in S9
(1
2 3 4 5 6 7 8 9
(1 2 3 4 5 6 7 8 9
and r=
(3
5 2 1 4 6 9 78
5 2 6 7 1 3 8 9
1) Write o,r, o
and
1' in cycle notation.
2) Write o and
as 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.
2/2
II
...
Transcribed Image Text:12:37 A M f Math365 Assign... / Exercise 4. Consider the following permutations in S9 (1 2 3 4 5 6 7 8 9 (1 2 3 4 5 6 7 8 9 and r= (3 5 2 1 4 6 9 78 5 2 6 7 1 3 8 9 1) Write o,r, o and 1' in cycle notation. 2) Write o and as 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. 2/2 II ...
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 9
5 4 3 29 8 67 6
a =
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.
Transcribed Image Text: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 9 5 4 3 29 8 67 6 a = 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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