1. ## Order of Permutation

Hello. Could someone help me with this?

Find the order of the permutation, sigma =
(1 2 3 4 5 6 7 8
3 1 5 6 2 7 8 4)
and write it as a product of cycles.

It's for a graduate school project and honestly I have never seen a problem like this before. So I appreciate any guidance.

2. Originally Posted by NYMFan
Hello. Could someone help me with this?

Find the order of the permutation, sigma =
(1 2 3 4 5 6 7 8
3 1 5 6 2 7 8 4)
and write it as a product of cycles.

It's for a graduate school project and honestly I have never seen a problem like this before. So I appreciate any guidance.
So you have the premutation on $\displaystyle \{1,2,3,4,5,6,7,8\}$ defined as,
$\displaystyle \sigma = \left( \begin{array}{cccccccc}1&2&3&4&5&6&7&8\\ 3&1&5&6&2&7&8&4 \end{array}\right)$
This is an element of a finite group $\displaystyle S_8$ thus it has an order. We need to find the smallest positive integer $\displaystyle n$ such that $\displaystyle \sigma^n$ is the identity premutation.
Observe the following,
$\displaystyle 3\to 5\to 2\to 1$
$\displaystyle 1\to 3\to 5\to 2$
$\displaystyle 5\to 2\to 1\to 3$
$\displaystyle 6\to 7\to 8\to 4$
$\displaystyle 2\to 1\to 3\to 5$
$\displaystyle 7\to 8\to 4\to 6$
$\displaystyle 8\to 4\to 6\to 7$
$\displaystyle 4\to 6\to 7\to 8$
This show that $\displaystyle n=3$ is the order.

To express it as a product of disjoint cycles we find the orbits of $\displaystyle \sigma$ which are,
$\displaystyle \{1,3,5,2\}$
$\displaystyle \{4,6,7,8\}$
Thus,
$\displaystyle \sigma=(1,3,5,2)(4,6,7,8)$
(Note although the binary operation on the premutation group is not commutative here it is and I could have expressed the product the other way aroung because the cycles are disjoint).