Proving the Order of the Symmetric Group S_N: Where to Begin?

[bjogae]

Homework Statement

Show that the symmetric group (permutation group) S_N is of order N!

Homework Equations

The Attempt at a Solution

I can't get started on how to prove this. I understand that if

n=2 S_2= {E, (12)}
n=3 S_3={E, (12), (13), (23), (123), (132)}

and so on. This makes this seem kind of intuitive. However I can't even get started on proving it for N. Could somebody please help me get started?

 

[VeeEight]

Given {1, 2, ..., n-1, n}, you want to construct a bijection onto itself. So take the first element, 1. There are n choices of elements it can map to. Then consider 2. There are n-1 elements it can map to (since the function is 1-1 and one of the n elements is already 'taken' by 1)

 

[bjogae]

Got it. Thanks.