Problems with permutations and transpositions

• morrowcosom
In summary, when trying to solve a problem that asks for the identity permutation, one can only use a product of transpositions that is even.
morrowcosom

Homework Statement

1)Consider the permutation in S3 = ( 1 2 3 )
( 1 2 3 ) NOTE: the two pairs of parenthesis are
meant to be one pair that encases both rows

Write as a product of transpositions

The Attempt at a Solution

(1 3) (1 2), which was wrong. I randomly plugged in numbers until I ended up with the correct solution of (1,3) (1,2) (1,3) which contradicts all the examples of transpositions I have seen, like (1 3 2 4) = (1 4) (1 2) (1 3).
How does one go about finding the solution to my problem?

morrowcosom said:

Homework Statement

1)Consider the permutation in S3 = ( 1 2 3 )
( 1 2 3 ) NOTE: the two pairs of parenthesis are
meant to be one pair that encases both rows
So
$$\begin{pmatrix} 1 & 2 & 3 \\ 1 & 2 & 3\end{pmatrix}$$
This is the identity permutation. Normally it is written just as "i". If you really want to write it as a product of transpositions, you could use (1 2)(1 2). That is, transpose 1 and 2, then transpose them again, going back to what you had originally. (1 3)(1 3) will also work as will (2 3)(2 3). Or could do two "transpositions and back": (1 3)(1 3)(1 3)(2 3)(2 3). Since each "transposition and back" requires two transpositions, no matter how many times we do that, we will have an even number of transpositions.

Write as a product of transpositions

The Attempt at a Solution

(1 3) (1 2), which was wrong. I randomly plugged in numbers until I ended up with the correct solution of (1,3) (1,2) (1,3) which contradicts all the examples of transpositions I have seen, like (1 3 2 4) = (1 4) (1 2) (1 3).
How does one go about finding the solution to my problem?
No, (1 3)(1 2)(1 3) will not work. (1 3) takes (1 2 3) to (3 2 1), then (1 2) take it to (3 1 2), and finally, (1 3) to (1 3 2), not (1 2 3). Since the identity permutation is an even permutation, it can only be written as a product of an even number of transpositons, not 3.

The identity transformation is pretty trivial. Are you sure you have copied the problem correctly?

No, (1 3)(1 2)(1 3) will not work. (1 3) takes (1 2 3) to (3 2 1), then (1 2) take it to (3 1 2), and finally, (1 3) to (1 3 2), not (1 2 3). Since the identity permutation is an even permutation, it can only be written as a product of an even number of transpositons, not 3.

When you say (1 3) takes (1 2 3) to (3 2 1), then (1 2) takes it to (3 1 2) could you explain the process of how this occurs? I am fine with multiplying permutatations and disjointed cycles, transposition confuses me though. Yes, the identity transformation was copied right.

I got this problem through cow.temple.edu. and am doing independent study.

Thanks

1. What is the difference between a permutation and a transposition?

A permutation is a rearrangement of a set of objects or elements, while a transposition is a specific type of permutation that only involves swapping two elements in a sequence.

2. How do problems with permutations and transpositions arise in real-world scenarios?

Problems with permutations and transpositions can arise in various fields such as computer science, mathematics, and genetics where ordering or rearranging elements is important. For example, in computer algorithms, rearranging data can affect the efficiency of the algorithm.

3. How do you calculate the number of possible permutations and transpositions?

The number of possible permutations can be calculated using the formula n! (n factorial), where n represents the number of elements in the set. For transpositions, the formula is n(n-1)/2.

4. Are there any strategies to solve problems with permutations and transpositions efficiently?

Yes, there are various strategies such as using mathematical formulas, creating algorithms, and using data structures like trees to deal with problems involving permutations and transpositions.

5. Can problems with permutations and transpositions be solved manually?

Yes, for smaller sets, problems with permutations and transpositions can be solved manually by listing out all the possible arrangements and counting them. However, for larger sets, it is more efficient to use mathematical formulas or algorithms.

• General Math
Replies
4
Views
1K
• Linear and Abstract Algebra
Replies
8
Views
2K
• Linear and Abstract Algebra
Replies
7
Views
1K
• Linear and Abstract Algebra
Replies
3
Views
1K
• Precalculus Mathematics Homework Help
Replies
32
Views
1K
• Precalculus Mathematics Homework Help
Replies
1
Views
861
• Precalculus Mathematics Homework Help
Replies
5
Views
2K
• Math Proof Training and Practice
Replies
23
Views
1K
• Precalculus Mathematics Homework Help
Replies
7
Views
2K
• Precalculus Mathematics Homework Help
Replies
23
Views
1K