Question on Permutations and Products of Transpositions.

In summary, the conversation discusses a given permutation and how to write it as a product of transpositions and determine its sign. The final conclusion is to determine if the permutation is written as an odd or even number of transpositions and what the corresponding sign would be.
  • #1
Wesc
12
1
Hi all, I've answered a question but there's no answer for it, and if ye could tell me if I'm doing it right I'd appreciate it thanks :)

Permutation: 1 2 3 4 5 6 7 8 9 10 11 12 13 14
--------------------2 3 1 6 5 4 8 10 13 11 12 7 14 9

(i) Write it as a product of transpositions.

I got the answer (1 2)(2 3)(4 6)(7 8)(8 10)(10 11)(11 12)(13 14)(14 9) ... But unsure if this is correct !

(ii) What is the sign of the permutation in terms of the product of transpositions? ... Not sure what the answer is here :( Could anyone show me how to do it?
 
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  • #2
Wesc said:
Hi all, I've answered a question but there's no answer for it, and if ye could tell me if I'm doing it right I'd appreciate it thanks :)

Permutation:
Code:
1 2 3 4 5 6 7 8  9  10 11 12 13 14
2 3 1 6 5 4 8 10 13 11 12 7  14 9

(i) Write it as a product of transpositions.

I got the answer (1 2)(2 3)(4 6)(7 8)(8 10)(10 11)(11 12)(13 14)(14 9) ... But unsure if this is correct !

Seems ok.

(ii) What is the sign of the permutation in terms of the product of transpositions? ... Not sure what the answer is here :( Could anyone show me how to do it?

Is it written as a odd or an even number of permutations? What is the corresponding sign?
 
  • #3
micromass said:
Seems ok.



Is it written as a odd or an even number of permutations? What is the corresponding sign?

Obviously >.< *Facepalm*

Thank you.
 

1. What is a permutation?

A permutation is a way of arranging a set of objects in a specific order. For example, if we have the set {1, 2, 3}, one permutation would be {1, 2, 3}, another would be {2, 1, 3}, and so on.

2. What is a transposition?

A transposition is a type of permutation that involves swapping two elements in a set. For example, if we have the set {1, 2, 3}, a transposition would be {2, 1, 3} where the 1 and 2 are swapped.

3. How do you calculate the number of possible permutations for a set?

The number of possible permutations for a set can be calculated using the factorial function. For a set with n elements, the number of permutations is n!. For example, if we have a set with 3 elements, there are 3! = 6 possible permutations.

4. What is the product of transpositions?

The product of transpositions refers to the result of performing multiple transpositions on a set. This can also be thought of as a composition of permutations. The order in which the transpositions are performed matters and can affect the final result.

5. How does the number of transpositions affect the product of transpositions?

The number of transpositions used in the product can affect the final result. If there is an even number of transpositions, the product will result in the original set. However, if there is an odd number of transpositions, the product will result in a different permutation of the original set.

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