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 should be 3, i.e., and for .
The order 3 subgroups of 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 is 4, which agrees to the third Sylow theorem, i.e., and .