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Proposition: The signature of any r-cycle in S, is equal to (-1)-. Proof: 1. Note that a typical r-cycle (aj a2 a,) E S, can be written as a product of 2-cycles as: ... 2. (a1 a2 a,) = (a, a,)(aj a,-1) ..* (a, a2). ... 3. We know that the signature of the r-cycle on the LHS is equal to the product of the 2-cycles on the RHS. 4. Note that there are 2-cycles on the RHS each of which has signature of 5. Therefore the signature of the r-cycle must be (-1)"-. a. Fill in the gaps in the proof. b. For a set S = {a1, .., an} consider the cycle f = (a1 a,). What properties does the cycle f have? ... O A. f has length r- 1. O B. f is a permutation of S. O C. f sends a1 → a2, ... , a,-1 → a, and a, → a1. O D. f is an onto (i.e. surjective) function on S. O E. f has a signature of -1 c. Which step of the proof used the fact that for any g, h E Sn, sgn(gh) sgn(g)sgn(h)? d. Where does the cycle on the LHS send a1? and where does the product of cycles on the RHS send a,? e. What is the signature of the cycle (1 2 3 4 5)?

Proposition: The signature of any r-cycle in S, is equal to (-1)-. Proof: 1. Note that a typical r-cycle (aj a2 a,) E S, can be written as a product of 2-cycles as: ... 2. (a1 a2 a,) = (a, a,)(aj a,-1) ..* (a, a2). ... 3. We know that the signature of the r-cycle on the LHS is equal to the product of the 2-cycles on the RHS. 4. Note that there are 2-cycles on the RHS each of which has signature of 5. Therefore the signature of the r-cycle must be (-1)"-. a. Fill in the gaps in the proof. b. For a set S = {a1, .., an} consider the cycle f = (a1 a,). What properties does the cycle f have? ... O A. f has length r- 1. O B. f is a permutation of S. O C. f sends a1 → a2, ... , a,-1 → a, and a, → a1. O D. f is an onto (i.e. surjective) function on S. O E. f has a signature of -1 c. Which step of the proof used the fact that for any g, h E Sn, sgn(gh) sgn(g)sgn(h)? d. Where does the cycle on the LHS send a1? and where does the product of cycles on the RHS send a,? e. What is the signature of the cycle (1 2 3 4 5)?

Advanced Engineering Mathematics
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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Proposition: The signature of any r-cycle in S, is equal to (-1)-. Proof: 1. Note that a typical r-cycle (aj a2 a,) E S, can be written as a product of 2-cycles as: 2. (aj a2 a,) = (aj a,)(aj a,-1) ·…· (aj az). 3. We know that the signature of the r-cycle on the LHS is equal to the product of the 2-cycles on the RHS. 4. Note that there are 2-cycles on the RHS each of which has signature of 5. Therefore the signature of the r-cycle must be (-1)"-'. a. Fill in the gaps in the proof. b. For a set S {a1, ... , an} consider the cycle f = (a1 ... a,). What properties does the cycle f have? O A. f has length r – 1. O B. f is a permutation of S. C. f sends aj → a2, .. ar-1 → a, and a, → aj . ... , O D. f is an onto (i.e. surjective) function on S. O E. f has a signature of –1 c. Which step of the proof used the fact that for any g, h E Sn, sgn(gh) = sgn(g)sgn(h)? d. Where does the cycle on the LHS send a1? and where does the product of cycles on the RHS send a,? e. What is the signature of the cycle (1 2 3 4 5)?
Transcribed Image Text:Proposition: The signature of any r-cycle in S, is equal to (-1)-. Proof: 1. Note that a typical r-cycle (aj a2 a,) E S, can be written as a product of 2-cycles as: 2. (aj a2 a,) = (aj a,)(aj a,-1) ·…· (aj az). 3. We know that the signature of the r-cycle on the LHS is equal to the product of the 2-cycles on the RHS. 4. Note that there are 2-cycles on the RHS each of which has signature of 5. Therefore the signature of the r-cycle must be (-1)"-'. a. Fill in the gaps in the proof. b. For a set S {a1, ... , an} consider the cycle f = (a1 ... a,). What properties does the cycle f have? O A. f has length r – 1. O B. f is a permutation of S. C. f sends aj → a2, .. ar-1 → a, and a, → aj . ... , O D. f is an onto (i.e. surjective) function on S. O E. f has a signature of –1 c. Which step of the proof used the fact that for any g, h E Sn, sgn(gh) = sgn(g)sgn(h)? d. Where does the cycle on the LHS send a1? and where does the product of cycles on the RHS send a,? e. What is the signature of the cycle (1 2 3 4 5)?
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