How do I express an element in a matrix of s8 as a product of disjoint cycles?

[beetle2]

Homework Statement

Express the element in matrix

A=

1,2,3,4,5,6,7,8
4,1,3,2,8,5,6,7

of s₈ as a product disjoint cycles

2. Homework Equations [/b

The Attempt at a Solution

I pick a number say the first 1 and pu it in parenthises. (1,
I multiply by the number in th image so number 1x 4 which gives 4 so that does not equal my first number 1 so I add it to my list so we now have (1,4

I than go to the next column which is number 2 and multiply that by the image so 2 x 1 that equals 2 which does not equal 1 so I add that to my list so now its (1,4,2

Now the next column has 3 and 3 in the image so I disregard and close the parenthises

(1,4,2)

Is this right so far I'm not sure if I'm using the right process.


regards

 

[micromass]

Ok, it's possible we don't mean the same thing here. But here I think it goes... (and it has nothing to do with multiplying)

1,2,3,4,5,6,7,8
4,1,3,2,8,5,6,7

Take the first element in the list, that's 1. Add it: it gives (1 ...
The element under 1 is 4, add that to the list: (1 4 ...
The element under 4 is 2, add that to the list: (1 4 2 ...
The element under 2 is 1, we have that already in our list, so we close the paranthesis: (1 4 2)

Take the next element not in our list, that's 3: (1 4 2)(3 ...
The element under 3 is 3, we have that already in our list, so we close the paranthesis: (1 4 2)(3)

Take the next element not in our list, that's 5: (1 4 2)(3)(5 ...
The element under 5 is 8, add that to the list: (1 4 2)(3)(5 8 ...
The element under 8 is 7, add that to the list: (1 4 2)(3)(5 8 7 ...
The element under 7 is 6, add that to the list: (1 4 2)(3)(5 8 7 6 ...
The element under 6 is 5, we have that already in our list, so we close the paranthesis: (1 4 2)(3)(5 8 7 6)

We have exhausted all the elements, so this is the final answer.

 

[beetle2]

I'm not sure. I checked then answers in the book and it has (142)(5876) not sure how they got it as theirs dosn't have a (3)?

 

[micromass]

Yes, both (142)(5876) as (142)(3)(5876) are correct. But it's a bit of a convention not to write the 1-cycles (like (3) ). So I guess (142)(5876) is the more standard solution. I should have mentioned that...

 

[beetle2]

thankyou very much for the explanation micromass much appreciated