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Let $G$ be the symmetric group of degree six. Identify a subgroup of $G$ of order 16 and decompose it as much as you can. This came up on and old exam. How would you do this without having access to listing of all the subgroups of $Sym(6)$

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You should be able to find a copy of the dihedral group (the group of symmetries of a square, an $8$-element group) inside the symmetric group on $4$ letters. With the two letters left over, you can form a transposition, and bring it up to a group of order $16$.

I'm not sure what's meant by "decompose it as much as you can," but from the construction I have outlined you should know enough about the group to answer that part of the question, as well.

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Hint: First by Lagrange Theorem G can contain elements with order 1,2,4,8,16. I'm sure that you have already known this, then you should condiser the possible orders for the elements of Sym(6), then you will recognize something. Ok this hint was just useful for you to recognize what kind of permutations you need for your group, now observed that order of Sym(6) is 720 which is 16.9.5, then Sym(6) has Sylow 2-subgroup of order 16. In addition, Sylow Theorem also states that all Sylow 2-subgroups are isomorphic. If you have checked the possible elements for G, you can write them as a direct product which is isomorphic to a well known group. Hope that this would be helpful.

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