Expert: Sherry Wallin - 5/27/2011

Question
Hi ma'am,

My question is: find the order of the element (1 2)(3 4 5) in S5. I'm struggling with how to find the elements of S5 so I can't even begin to find the order.

Any help you could offer would be greatly appreciated!

Thank you,
Nik

Answer
I want to add a couple of notes below my signature

Nik~

The elements in S5 are composed of all possibilities of the digits 1,2,3,4, and 5
Use counting theory to determine how many positions can 1 be in?  Yes 5, and then how many places can 2 be in? Yes 4, and then how many places can 3 be in? Yes 3, and how many places can 4 be in Yes 2, and how many choices are there for the 5th position? Yes just 1. So in S5 there are
5! = 5*4*3*2*1 = 120 elements which is the order of S5.

I will list a few of the elements of S5: (1 2 3 4 5), (1 3 4 5 2), (1 2 4 5 3), ...

Every permutation is a product of disjoint cycles. Also, the order of a product
of disjoint cycles is the least common multiple of the orders of the cycles. For example the
permutation (123)(4567)(89) has order lcm(3, 4, 2) = 12. Thus the order of (12)(345) is lcm(2,3) = 6

Math Prof

The reason that the order is 6 (besides the fact that the lcm(2,3) = 6) is because you need to raise (1 2) to the 2nd power to get the identity and you need to raise (3 4 5) to the 3rd power to get the identity. I will show you a way to calculate these and try to explain what I am doing.

Let's start with (1 2) and (1 2) and we let the first (1 2) = f and the 2nd (1 2) = g.
    f         g
1----> 2 ----> 1
see you are already at f's identity. I will also do (3 4 5)(3 4 5)(3 4 5), letting the 1st (3 4 5) = f, the 2nd (3 4 5) = g, and the
3rd (3 4 5) = h


    f         g
3----> 4---->5
    f         g
5----> 3---->4
    f         g
4----> 5---->3

thus (3 4 5) (3 4 5) = (5 4 3) = fg

so we want (3 4 5)^2 *(3 4 5) = fg * h =>

(5 4 3) (3 4 5) =>

    fg        h
5----> 4---->5  => we now have the identity

Thus the order is the power you need to raise your first element to in order to get the identity and the power you need to raise your second element to to get it's identity, i.e., 2*3 = 6