Conjugating Sylow 3-subgroups in S_4

[ehrenfest]

[SOLVED] Sylow theory

Homework Statement

Find all Sylow 3-subgroups of S_4 and demonstrate that they are all conjugate.

Homework Equations

The Attempt at a Solution

I found all the Sylow 3-subgroups, but I am having trouble finding the element that conjugates them. For example, how do you find the element conjugates that <(1,2,3)> and <(2,3,4)>? I could just try all the elements, but there are 24, so that is probably a bad idea...

 

[jacobrhcp]

well, you can just try all elements, leave a few by symmetry.

But you'd only have to do them all once, because if you did it correct you'll see all the sylow subgroups emerge =)

If choose carefully you'll probably only have to do a bit more than there are subgroups in the class.

 

[NateTG]

You could also try to solve the general case:
$$p (1, 2, 3) p^{-1} = (a,b,c)$$
with
[tex]a\neq b \neq c[/itex][/tex]