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I want to find a $2$-Sylowgroup, i.e. a subgroup of order $8$, of $A_6$.

I think I can solve this problem by brute force but in order to avoid endless calculation I was wondering whether someone could kindly provide a more systematic or efficient approach or any theorems that might help for this question.

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    Relevant. – Shaun Apr 09 '22 at 19:05
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    Check out this older thread. I don't know if it is at the level you seek, but my guess is that it would be close. Once you have a Sylow 2-subgroup of $S_n$ you get one for $A_n$ simply by only including the even permutations. – Jyrki Lahtonen Apr 09 '22 at 19:05
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    There’s an injective homomorphism $S_4\to A_6$ that is the identity on $A_4$ and is defined on odd elements by $\sigma\mapsto \sigma(56)$ and there is a natural embedding $D_8\to S_4$. Combine these maps to get the Sylow subgroup you’re looking for. – David Hill Apr 09 '22 at 20:04

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