6. Show that a cycle of odd length is an even permutation, and a cycle of even length is an oddpermutation. Hence determine which of the following members of S10 are even, or odd:(i) (19 3) (2 6) (4 5 10 ) (78)1 9 3 2 6) (4 5 10 7 8),(iii) 1 9 3 2 6 4 5 10).

Answered: Image /qna-images/answer/11aefcb7-9323-4281-9fad-40fc0f6e03be.jpg

Answered: . Show that a cycle of odd length is an…

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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. (a) If g is a primitive root modulo 37, which of the numbers g², g³,..., g8 are primitive roots modulo 37?
Is there a formula or way to determine how many transpositions there are for a cycle of a given length? Is there a bound of sorts on the minimum and maximum number of transpositions for a cycle of a given length? For example, suppose I have a cycle of length 5. I know there are only an even number of transpositions possible, but how many are there actually? Wouldn't the smallest number be 4 transpositions for a cycle of length 5? I know there is one with 6 transpositions possible for a cycle of length 5? Are there others? If so, can we determine them? Is it correct to say that if a permutation is a 5-cycle, then it has 2s number of possible transpositions?
I. Consider the permutation (1 2 3)(4 5)(1 2 5 4) ~². 1. Express as a product of disjoint cycles. 2. What is the order of the given permutation? 3. Express your answer in number 1 as a product of transpositions. 4. Determine whether the permutation is even or odd.
n2 +1 is of the form Problem 4. Let n E N, prove that every prime divisor p of the number n4 12k +1 for some integer k.
(a) Write (152) (341) (6352) ≤ S6 in array form. Fill in the blank row below. 1 2 3 4 5 6 (b) Express [ 1 2 3 4 5 6 7 8 9 1 4 2 97 8 5 3 6 as a product of disjoint cycles. (c) How many elements of order 10 does S7 have?
6. Discover the recursive formula for Hn. Find Hn for n=1,2,3...6. Based on these values you will be able tosee the recursive pattern. proving the formula is optional for extra credit. Consider all subsets of N={1, 2, 3,..., n). How many of these subsets, SC N, satisfy the property that (i) Vx (x ES → x +1 ¢ S) ? Denote the number of subsets satisfying this property (i) by Hn. (a) Find the first 6 values of Hn . I have calculated n=1 and n=2, so you need to do n=3, 4, 5, 6 (b) Find and prove the recursive formula for the sequence Hn . Example: Let n=1. Then N={1} and the subsets satisfying property (i) are {1} and O so H1=2. Let n=2. Then N={1,2} and the subsets satisfying property (i) are {1}, {2}, and Ø, i.e. H3=3. Note: {1,2} is also subset of {1,2} but since for x=1, x+1=2 and both 1 and 2 are contained in {1,2} therefore the set {1,2} fails to satisfy property (i). 7. Prove by induction that the number of subsets for a set with n elements is 2^n.

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. Show that a cycle of odd length is an even permutation, and a cycle of even length is an odd permutation. Hence determine which of the following members of S10 are even, or odd: 10) (78) (i) (19 3) ( 6) (45 (ii) (1 9 3 2 6 ) (4 5 10 7 8), (iii) (1 9 3 2 6 4 5 10).