Sylow 3-subgroups of symmetric group

Question

Is there any routine technique to find a set of permutations which generate a Sylow $3$-subgroup of the symmetric group $S_{n}$?

Answer 1

Such a Sylow $3$-subgroup is a direct product of "iterated wreath products" of the cyclic group of order $3$. It depends on the base $3$ expansion of $n.$ If $n = a_{0} + 3 a_{1} + \ldots + 3^{m-1}a_{m-1}$ where each $a_{i} \in \{0,2 \}$, then a Sylow $3$-subgroup of $S_{n}$ is the direct product over $i$ of a direct product of $a_{i}$ copies the same as a Sylow $3$-subgroup of $S_{3^{i}}$. This reduces us to considering the case $n = 3^{m}$ for some $m$.

In that case, one takes a direct product of $3$ copies of a Sylow $3$-subgroup of $S_{3^{m-1}},$ and then adjoins a product of $3^{m-1}$ $3$-cycles interchanging the sets of points moved by the respective direct factors in a way compatible with their respective actions. This is more complicated to say than to implement: for example, to go from $S_{3}$ to $S_{9},$ take $\langle (123) \rangle \times \langle (456) \rangle \times \langle (789) \rangle$ and adjoin $(147)(258)(369).$

There is nothing in this construction which is particular to the prime $3$. The obvious analogues work for Sylow $p$-subgroups of $S_{n}$ for any prime $p.$

Answer 2

Answer by example:
Let's take $n=16=9+3+3+1$.

The 3-Sylow of $S_n$ is a subgroup of the symmety group of this graph: $$ \bullet\!\!\!\stackrel{\textstyle/}{\phantom{\bullet}}\stackrel{\textstyle\bullet}{\stackrel{\textstyle |}\bullet}\stackrel{\textstyle\backslash}{\phantom{\bullet}} \!\!\!\!\bullet \!\!\! \begin{matrix}\diagup\\\phantom{\Big|}\\\phantom{.}\\\phantom{.}\\\end{matrix} \!\! \bullet\!\!\!\stackrel{\textstyle/}{\phantom{\bullet}} \stackrel{\textstyle\bullet}{\stackrel{\textstyle |} {\stackrel{\textstyle\bullet}{\stackrel{\textstyle |}\bullet}}}\stackrel{\textstyle\backslash}{\phantom{\bullet}} \!\!\!\!\bullet \!\! \begin{matrix} \diagdown\\\phantom{.}\\\phantom{\Big|}\\\phantom{.}\\ \end{matrix} \!\!\! \bullet\!\!\!\stackrel{\textstyle/}{\phantom{\bullet}}\stackrel{\textstyle\bullet}{\stackrel{\textstyle |}\bullet}\stackrel{\textstyle\backslash}{\phantom{\bullet}} \!\!\!\!\bullet \quad \bullet\!\!\!\stackrel{\textstyle/}{\phantom{\bullet}}\stackrel{\textstyle\bullet}{\stackrel{\textstyle |}\bullet}\stackrel{\textstyle\backslash}{\phantom{\bullet}} \!\!\!\!\bullet\,\,\,\,\, \bullet\!\!\!\stackrel{\textstyle/}{\phantom{\bullet}}\stackrel{\textstyle\bullet}{\stackrel{\textstyle |}\bullet}\stackrel{\textstyle\backslash}{\phantom{\bullet}} \!\!\!\!\bullet\quad \bullet $$ It consists of those symmetries that (1) do not permule the connected components, and (2) preserve the cyclic order among the children of each vertex.

A minimal set of generators is given by: $$ (1,2,3),\quad (1,4,7)(2,5,8)(3,6,9),\quad (10,11,12),\quad (13,14,15). $$