Understanding Even Permutations in Abstract Algebra

In summary, A permutation can be written as a product of transpositions and there are multiple ways to do so. However, all the ways will have the same parity, either odd or even. To determine if a permutation is odd or even, you can count the number of transpositions in any factorization. If it is odd, then the permutation is odd, and if it is even, then the permutation is even. Additionally, if a permutation is written as a product of disjoint cycles, you can determine its parity by counting the lengths of the cycles and subtracting the number of cycles from the sum of the lengths.
  • #1
Bellarosa
48
0
1. This is not a question it's an example.



2.The permutation (123)= (13)(12)= (13)(23)(12(13)= (23)(13)(12)(13)(12)(23) is even.




3. I got the frist one because it is the product of tranposition...I just don't get the rest. I know that it is even depending on the number of transposition it contains, getting them is my problem.
 
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  • #2
what is your question?
There is a theorem that says, that every permutation can be written as a product of transpositions.

[tex](d_1d_2...d_n)=(d_nd_1)(d_nd_2)...(d_nd_{n-1})[/tex]

But you have to understand that there are many ways of rewriting a permutation as a product of transpositions, i.e

[tex]\theta=\alpha_1\alpha_2...\alpha_r=\beta_1\beta_2...\beta_s=\delta_1\delta_2...\delta_m[/tex]

but all r,s and m have the same parity, that is they are either odd or even.(there is a theorem that establishesh this fact as well).
 
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  • #3
The "rest" of what? There exist an infinite number of ways any permutation can be written as a product of transpositions.
 
  • #4
want to figure out whether a permutation is odd or even...my textbook gave the example (123) = (12)(13), (123) = (13)(23)(12)(13), (123)= (23)(13)(12)(13)(12)(23) they called it the factorization as a product of transformations.
Now I see that (123) is even because for each factorization there is an even number of transposition, what I don't get is how to derive the factorization of transformations.

Here is another example:(1235) = (15)(24)(24)(13)(23)(23)(12), (1235) = (13)(24)(35)(14)(24), (1235) = (15)(13)(12)...
...I'm trying to determine whether (12) (134) (152) this is even or odd, In the examples above I only hadone permutation , now I have cycles...
 
  • #5
(1235) means "1 becomes 2, 2 becomes 3, 3 becomes 5, 5 becomes 1 and all others remain the same": (12345) is changed to (23541).

You can write that as transpositions by: (12)(13)(15)- I got that by thinking "first 1 and 2 swap. Now that 1 is in the old "2" place, it swaps with 3, once it is the "3" place it swaps with 5: first (12345) becomes (21345) then (23145) then (23541). That is your third option. Because that is 3 (an odd number of) transpositions this is an odd permutation. But notice that all of your possible factorizations has an odd number of transpositions: 7, 5, and 3.

To factor (12)(134)(152), note that the first (12) is already a transposition. (134) can be factored as (13)(14) and (152) as (15)(12). (12)(134)(152)= (12)(13)(14)(15)(12) which has 5 transpositions- this is an odd permutation.
 
  • #6
In case you have a permutation written as a product of say t disjoint cycles, then you can tell whether the permutation is odd or even by just counting the lengths of each cycle and subtracting from it the number of cycles. That is say

[tex]\theta=\alpha_1\alpha_2\alpha_3...\alpha_t[/tex] are t disjoint cycles each with length [tex]s_1,s_2,...,s_t[/tex] then this permutation is odd or even depending on whether the following is odd or even

[tex]s_1+s_2+...+s_t-t[/tex]
 

1. What is an even permutation?

An even permutation is a type of permutation in abstract algebra that can be represented as a product of an even number of transpositions. This means that it can be obtained by swapping two elements in a sequence an even number of times. In other words, the parity of the permutation is even.

2. How are even permutations represented in abstract algebra?

In abstract algebra, even permutations are usually represented using the notation (12)(34)(56), where the numbers in the parentheses represent the elements being swapped. This notation is known as cycle notation and is a compact way of representing permutations.

3. What is the significance of even permutations in abstract algebra?

Even permutations are important in abstract algebra because they form a subgroup of the symmetric group, which is the group of all possible permutations. This subgroup is known as the alternating group and has many important applications in areas such as group theory, combinatorics, and cryptography.

4. How are even permutations related to odd permutations?

Odd permutations are the ones that cannot be represented as a product of an even number of transpositions. This means that their parity is odd. In abstract algebra, odd permutations are the elements that are not part of the alternating group. They are also important in understanding the structure of the symmetric group.

5. Can even permutations be applied to real-life scenarios?

Yes, even permutations have many real-life applications, especially in cryptography. They are used in encryption algorithms to scramble data in a way that can only be reversed by applying the same even permutation in the correct order. This makes them an important tool for securing sensitive information.

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