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Is there any routine technique to find a set of permutations which generate a Sylow 2-subgroup of the symmetric group $S_{2^{r−1}}$?

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up vote 2 down vote accepted

The Sylow $2$-subgroup of $S_n$, where $n$ is a power of $2$, is the automorphism group of a perfect binary tree with $n$ leaves. So, by induction, you can generate the Sylow $2$-subgroup by generators of $S_{n/2}$, plus anything which swaps the children of the root:

$\langle (1~2), \\\ (1~3)(2~4),\\\ (1~5)(2~6)(3~7)(4~8),\\\ ..., \\\ (1~~1+n/2)(2~~2+n/2)(3~~3+n/2)...(n/2~~n) \rangle$

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Yes. There is a method to find $p$-Sylow subgroups of $S_n$, by wreath product. See page 18 of the great book "Permutation groups, Dixon and Mortimer".

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