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?
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 $\displaystyle 4=1\cdot3 + 1$.