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Well my queston is the title exactly. Im trying to get an isomorphism between a 5-sylow subgroup of alternating group $A_{20}$ and something familiar. But I don't know where to start. I know that $A_{20}$ is simple so I can't have a unique 5-sylow correct? and its order must be $5^{4}$.

Any help is appreciated. Thanks

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I should also say you are exactly correct: it cannot have a unique Sylow 5-subgroup, in fact it has 633568231296 of them. The order of every single one of them is $5^4 = \color{red}{5}\cdot \color{purple}{5}\cdot\color{green}{5} \cdot \color{blue}{5}$. – Jack Schmidt Nov 3 '11 at 21:40
Im sorry I dont understand this. Why does P look like this, and how does that make it isomorphic to C_{4} x C_{4} x C_{4} x C_{4}? – alice Nov 3 '11 at 22:48

You can write down a Sylow 5-subgroup pretty easily. Divide 1 to 20 into four colors: red is 1 to 5, purple is 6 to 10, green is 11 to 15, and blue is 16 to 20. Now you just take Sylow 5-subgroups for each color:

$$P = \langle \color{red}{(1,2,3,4,5)}, \color{purple}{(6,7,8,9,10)}, \color{green}{(11,12,13,14,15)}, \color{blue}{(16,17,18,19,20)} \rangle$$

This is the same as: $$ P \cong \color{red}{C_5} \times \color{purple}{C_5} \times \color{green}{C_5} \times \color{blue}{C_5}$$

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I didn't know color worked here! And also nice answer :) – user641 Nov 3 '11 at 22:39

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