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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.

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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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.
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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