Editing Sylow theorems (section)

== Examples ==
[[File:Labeled Triangle Reflections.svg|thumb|In ''D''<sub>6</sub> all reflections are conjugate, as reflections correspond to Sylow 2-subgroups.]]

A simple illustration of Sylow subgroups and the Sylow theorems are the [[dihedral group]] of the ''n''-gon, ''D''<sub>2''n''</sub>. For ''n'' odd, 2 = 2<sup>1</sup> is the highest power of 2 dividing the order, and thus subgroups of order 2 are Sylow subgroups. These are the groups generated by a reflection, of which there are ''n'', and they are all conjugate under rotations; geometrically the axes of symmetry pass through a vertex and a side.

[[File:Hexagon reflections.svg|thumb|left|In ''D''<sub>12</sub> reflections no longer correspond to Sylow 2-subgroups, and fall into two conjugacy classes.]]
By contrast, if ''n'' is even, then 4 divides the order of the group, and the subgroups of order 2 are no longer Sylow subgroups, and in fact they fall into two conjugacy classes, geometrically according to whether they pass through two vertices or two faces. These are related by an [[outer automorphism]], which can be represented by rotation through π/''n'', half the minimal rotation in the dihedral group.

{{Clear}}
Another example are the Sylow p-subgroups of ''GL''<sub>2</sub>(''F''<sub>''q''</sub>), where ''p'' and ''q'' are primes&nbsp;≥&nbsp;3 and {{math|''p''&nbsp;≡&nbsp;1&nbsp;(mod&nbsp;''q'')}} , which are all [[Abelian group|abelian]]. The order of ''GL''<sub>2</sub>(''F''<sub>''q''</sub>) is {{math|1=(''q''<sup>2</sup>&nbsp;−&nbsp;1)(''q''<sup>2</sup>&nbsp;−&nbsp;''q'') = (''q'')(''q''&nbsp;+&nbsp;1)(''q''&nbsp;−&nbsp;1)<sup>2</sup>}}. Since {{math|1=''q''&nbsp;=&nbsp;''p''<sup>''n''</sup>''m''&nbsp;+&nbsp;1}}, the order of {{math|1=''GL''<sub>2</sub>(''F''<sub>''q''</sub>) =&nbsp;''p''<sup>2''n''</sup> ''m''&prime;}}. Thus by Theorem&nbsp;1, the order of the Sylow ''p''-subgroups is ''p''<sup>2''n''</sup>.

One such subgroup ''P'', is the set of diagonal matrices  <math>\begin{bmatrix}x^{im} & 0 \\0 & x^{jm} \end{bmatrix}</math>, ''x'' is any [[Primitive root modulo n|primitive root]] of  ''F''<sub>''q''</sub>. Since the order of ''F''<sub>''q''</sub> is {{math|1=''q''&nbsp;−&nbsp;1}}, its primitive roots have order ''q'' − 1, which implies that {{math|1=''x''<sup>(''q''&nbsp;−&nbsp;1)/''p''<sup>''n''</sup></sup>}} or ''x''<sup>''m''</sup> and all its powers have an order which is a power of&nbsp;''p''. So, ''P'' is a subgroup where all its elements have orders which are powers of&nbsp;''p''. There are ''p<sup>n</sup>'' choices for both ''a'' and ''b'', making {{math|1 = {{!}}''P''{{!}} =&nbsp;''p''<sup>2''n''</sup>}}. This means ''P'' is a Sylow ''p''-subgroup, which is abelian, as all diagonal matrices commute, and because Theorem 2 states that all Sylow ''p''-subgroups are conjugate to each other, the Sylow ''p''-subgroups of ''GL''<sub>2</sub>(''F''<sub>''q''</sub>) are all abelian.