[FONT="] Find all the 3-Sylow subgroups of S4. I am having a real hard time getting the idea behind Sylow subgroups. Can someone help me with this please?[/FONT]

I'll assume S4 as the symmetric group of degree 4.

The possible order of a 3-Sylow subgroup of \(\displaystyle S_4\) should be 3, i.e., \(\displaystyle 3^1 \mid 24\) and \(\displaystyle 3^k \nmid 24\) for \(\displaystyle k \geq 2\).

The order 3 subgroups of \(\displaystyle S_4\) are simply the cyclic groups of order 3.

<(1 2 3)>, <(1 3 4)>, <(2 3 4)>, and <(1 2 4)>.

The number of 3-Sylow subgroups of \(\displaystyle S_4\) is 4, which agrees to the third Sylow theorem, i.e., \(\displaystyle 4 \mid 24\) and \(\displaystyle 4=1\cdot3 + 1\).