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)?
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.3: The Addition And Subtraction Formulas
Problem 79E
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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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