Solving Computations w/ Cycles: Permutations, Inverses, Orders

In summary, the conversation discussed finding the product and inverse of a permutation in S9. The correct answer for the product was (1 3 5 8 9)(2 6 4)(7), and the method for finding the inverse was mentioned as going backwards. The convention of not mentioning characters that are not changed by a permutation was also brought up.
  • #1
CoachBryan
14
0
So I've been sitting here for a while looking at my study guide and I am not sure how to find the product (or even the inverse) of this permutation in S9:

(2 5 1 3 6 4) (8 5 6)(1 9) = (1 3 8 5 9)(2 6 4) (Correct answer)

I know it starts off with 1 --> 3, then you get (1 3 and then after you continue with 3 --> 6 --> 8. After you start with 6 --> 4, right? But I keep coming across the wrong answer. I get (1 3 8 4 2) (8 1)

I'm pretty confused on this topic. If you could shed some light I'd greatly appreciate it. Thanks!
 
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  • #2
Well, it was a long time ago I did those things, but, assuming that the permutations should be performed from left to right, I get:

1 --> 3 --> 3 --> 3
3 --> 6 --> 8 --> 8
8 --> 8 --> 5 --> 5
5 --> 1 --> 1 --> 9
9 --> 9 --> 9 --> 1

leading to the cycle (1 3 8 5 9)

2 --> 5 --> 6 --> 6
6 --> 4 --> 4 --> 4
4 --> 2 --> 2 --> 2

leading to the cycle (2 6 4)

7 --> 7 --> 7 --> 7

leading to the cycle (7)

So, the answer is (1 3 5 8 9)(2 6 4)(7)

but (7) needs perhaps not be included, or maybe the elements to be permuted don't even include 7. In any case, the book's answer is correct.
 
  • #3
Erland said:
Well, it was a long time ago I did those things, but, assuming that the permutations should be performed from left to right, I get:

1 --> 3 --> 3 --> 3
3 --> 6 --> 8 --> 8
8 --> 8 --> 5 --> 5
5 --> 1 --> 1 --> 9
9 --> 9 --> 9 --> 1

leading to the cycle (1 3 8 5 9)

2 --> 5 --> 6 --> 6
6 --> 4 --> 4 --> 4
4 --> 2 --> 2 --> 2

leading to the cycle (2 6 4)

7 --> 7 --> 7 --> 7

leading to the cycle (7)

So, the answer is (1 3 5 8 9)(2 6 4)(7)

but (7) needs perhaps not be included, or maybe the elements to be permuted don't even include 7. In any case, the book's answer is correct.
Thanks a lot! Now it makes sense. Do you know how to find the inverse?
 
  • #4
CoachBryan said:
Thanks a lot! Now it makes sense. Do you know how to find the inverse?
Just go backwards!
 
  • #5
The convention that I am familiar with is that if a character is not changed by a permutation, it doesn't need to be mentioned.
So your (1 3 5 8 9)(2 6 4)(7) can be written as (1 3 5 8 9)(2 6 4).
 
  • #6
Thanks guys! On the inverse problem as well. Appreciate it.
 

FAQ: Solving Computations w/ Cycles: Permutations, Inverses, Orders

What is a permutation in computing?

A permutation in computing is a rearrangement of a set of objects or elements in a specific order. In programming, this can be represented as the arrangement of characters in a string or the arrangement of elements in an array.

How do you calculate the order of a permutation?

The order of a permutation can be calculated by finding the least common multiple of the lengths of all the cycles in the permutation. This can be done by breaking down the permutation into cycles and finding the length of each cycle.

What is an inverse in computing?

In computing, an inverse is the opposite or reverse of an operation. In the context of permutations, the inverse of a permutation is the rearrangement of the elements in reverse order.

How do you find the inverse of a permutation?

To find the inverse of a permutation, you can simply reverse the order of elements in the permutation. Alternatively, you can use the cycle notation and switch the direction of each cycle.

How are permutations and inverses used in computing?

Permutations and inverses are commonly used in encryption algorithms, such as the RSA algorithm, to securely rearrange data. They are also used in data compression and error-correcting codes where ordering and rearranging data can improve efficiency and accuracy.

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