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?

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- May 10th 2010, 06:31 PMnhkSubgroups of S4
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?

- May 12th 2010, 02:31 AMTheArtofSymmetry
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 .