Order of Permutations (1 2 3 4 5 6 7): Explained

[math8]

I am sure this is very simple but I m kind of confused here.

What is this product equal to and what's the order of the permutation?

(1 2 3 4 5 6 7) (3 6 7 4 2 5 1)

I thought it was (3 7 5 2 6 1 4) but I am reading somewhere that it should be (137)(265)(4) and hence has order 3.

Why is this "(3 7 5 2 6 1 4)" not correct? I mean I thought I had to start with the last cycle and do 3-->6, in the first cycle, 6-->7
and then 7-->4, in the first cycle, 4-->5,...

So I would get something like (3 7 5 ...

I am very confused here, help

 

[Dick]

That isn't cycle notation. Are you sure they didn't write this on two rows, like
(1 2 3 4 5 6 7)
(3 6 7 4 2 5 1)
The first row is just the argument of the permutation and the second row is it's value.
I.e. 1->3 2->6 3->7 4->4 5->2 6->5 7->1

 

[math8]

Oh I see, so if it was a product of cycles, my answer would have been correct and the order of the product would be 7. But in this case, we just have the argument of a permutation and its value. So the question here is not to find the order of the product of two cycles but rather the order of the single permutation (given the argument and the value).
Is that correct?

 

[Dick]

Oh I see, so if it was a product of cycles, my answer would have been correct and the order of the product would be 7. But in this case, we just have the argument of a permutation and its value. So the question here is not to find the order of the product of two cycles but rather the order of the single permutation (given the argument and the value).
Is that correct?

It looks like that to me. It seems to fit with the given answer.

 

[math8]

thanks.