# Subgroups of S4

#### nhk

[FONT=&quot] 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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#### TheArtofSymmetry

[FONT=&quot] 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 $$\displaystyle 4=1\cdot3 + 1$$.

nhk