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Find "all" the Sylow subgroups of S5 and S6
I don't know how to find 1.Sylow 2-subgroups of S5
2.Sylow 2-subgroups of S6
3.Sylow 3-subgroups of S6
4.Sylow 5-subgroups of S6
I use Sylow thms to get the possible numbers of Sylow subgroups but don't know how to find the right one.
Even I know the right number.
How to find them all?
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Sylow's theorems tell you that any two Sylow p-subgroups of a group G are conjugate in G so if you can find one it shouldn't be too hard to find them all. (I assume that "all" in quotation marks is acknowledging this.)
Sylow's theorems also tell you that any p-subgroup of G is contained in a Sylow p-subgroup of G, so in these small cases it shouldn't be too hard to find a p-subgroup and if it is too small find a bigger one containing it.
Can you determine the orders of these Sylow p-subgroups?
For instance a Sylow 5-subgroup of S6 must have order 5. Can you find a subgroup of order 5 in S6?
Finding a Sylow 3-subgroup of S6 isn't too much harder.
To find a Sylow 2-subgroup of S5 consider what its order has to be, and also what the order of a Sylow 2-subgroup of S4 must be.
We can naturally embed S4 in S5 so any Sylow 2-subgroup of S4 must be contained in a Sylow 2-subgroup of S5.
If you have found a Sylow 2-subgroup of S5, then you want to find a Sylow 2-subgroup of S6 containing it.
Consider what the order of such a subgroup should be and I don't think it is too hard to find.
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