Editing Symmetric group (section)

=== Sylow subgroups ===
The [[Sylow subgroup]]s of the symmetric groups are important examples of [[p-group|''p''-groups]].  They are more easily described in special cases first:

The Sylow ''p''-subgroups of the symmetric group of degree ''p'' are just the cyclic subgroups generated by ''p''-cycles.  There are {{math|1=(''p'' − 1)!/(''p'' − 1) = (''p'' − 2)!}} such subgroups simply by counting [[Presentation of a group|generators]].  The [[normalizer]] therefore has order {{math|''p''⋅(''p'' − 1)}} and is known as a [[Frobenius group]] {{math|''F''<sub>''p''(''p''−1)</sub>}} (especially for {{math|1=''p'' = 5}}), and is the [[affine general linear group]], {{math|AGL(1, ''p'')}}.

The Sylow ''p''-subgroups of the symmetric group of degree ''p''<sup>2</sup> are the [[wreath product]] of two cyclic groups of order ''p''.  For instance, when {{math|1=''p'' = 3}}, a Sylow 3-subgroup of Sym(9) is generated by {{math|1=''a'' = (1 4 7)(2 5 8)(3 6 9)}} and the elements {{math|1=''x'' = (1 2 3), ''y'' = (4 5 6), ''z'' = (7 8 9)}}<!-- or just use (1,2,3) -->, and every element of the Sylow 3-subgroup has the form {{math|1=''a''<sup>''i''</sup>''x''<sup>''j''</sup>''y''<sup>''k''</sup>''z''<sup>''l''</sup>}} for {{tmath|1=0 \le i,j,k,l  \le 2}}.

The Sylow ''p''-subgroups of the symmetric group of degree ''p''<sup>''n''</sup> are sometimes denoted W<sub>''p''</sub>(''n''), and using this notation one has that {{math|W<sub>''p''</sub>(''n'' + 1)}} is the wreath product of W<sub>''p''</sub>(''n'') and W<sub>''p''</sub>(1).

In general, the Sylow ''p''-subgroups of the symmetric group of degree ''n'' are a direct product of ''a''<sub>''i''</sub> copies of W<sub>''p''</sub>(''i''), where {{math|1= 0 ≤ ''a<sub>i</sub>'' ≤ ''p'' − 1}} and {{math|1=''n'' = ''a''<sub>0</sub>&nbsp;+&nbsp;''p''⋅''a''<sub>1</sub>&nbsp;+&nbsp;...&nbsp;+&nbsp;''p''<sup>''k''</sup>⋅''a''<sub>''k''</sub>}} (the base ''p'' expansion of ''n'').

For instance, {{math|1=W<sub>2</sub>(1) =&nbsp;C<sub>2</sub> and W<sub>2</sub>(2) =&nbsp;D<sub>8</sub>}}, the [[dihedral group of order 8]], and so a Sylow 2-subgroup of the [[symmetric group]] of degree 7 is generated by {{math|{ (1,3)(2,4), (1,2), (3,4), (5,6) } }} and is isomorphic to {{math|D<sub>8</sub> × C<sub>2</sub>}}.

These calculations are attributed to {{harv|Kaloujnine|1948}} and described in more detail in {{harv|Rotman|1995|p=176}}.  Note however that {{harv|Kerber|1971|p=26}} attributes the result to an 1844 work of [[Augustin-Louis Cauchy|Cauchy]], and mentions that it is even covered in textbook form in {{harv|Netto|1882|loc=§39–40}}.