Exercise 14.6.10. Prove that it is impossible to write the identity permu- tation as the product of an odd number of transpositions.
Exercise 14.6.10. Prove that it is impossible to write the identity permu- tation as the product of an odd number of transpositions.
College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter9: Counting And Probability
Section9.1: Counting
Problem 4E: True or False? In counting combinations, order matters. In counting permutations, order matters. For...
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Please do Exerise 14.6.10 please show step by step and explain
![Now here's the punch line. We know that every permutation can be writ-
ten as a product of transpositions. From what we have just shown, an odd
permutation must be the product of an odd number of transpositions; while
an even permutation must be the product of an even number of transposi-
tions. It is impossible to write an even permutation as the product of an odd
number of transpositions; and vice versa. We summarize our conclusions in
the following proposition.
Proposition 14.6.9. A permutation o can be written as the product of an
even number of transpositions if and only if o is an even permutation. Also,
o can be written as the product of an odd number of transpositions if and
only if o is an odd permutation.
Exercise 14.6.10. Prove that it is impossible to write the identity permu-
tation as the product of an odd number of transpositions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F892e817a-9b32-4eeb-b8fc-5dd7ffde6479%2F71c47b11-9dbf-43d4-afbd-ad44c0b5b751%2F4i5ffvd_processed.png&w=3840&q=75)
Transcribed Image Text:Now here's the punch line. We know that every permutation can be writ-
ten as a product of transpositions. From what we have just shown, an odd
permutation must be the product of an odd number of transpositions; while
an even permutation must be the product of an even number of transposi-
tions. It is impossible to write an even permutation as the product of an odd
number of transpositions; and vice versa. We summarize our conclusions in
the following proposition.
Proposition 14.6.9. A permutation o can be written as the product of an
even number of transpositions if and only if o is an even permutation. Also,
o can be written as the product of an odd number of transpositions if and
only if o is an odd permutation.
Exercise 14.6.10. Prove that it is impossible to write the identity permu-
tation as the product of an odd number of transpositions.
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