Editing Dihedral group (section)

===Conjugacy classes of reflections===
All the reflections are [[conjugacy class|conjugate]] to each other whenever ''n'' is odd, but they fall into two conjugacy classes if ''n'' is even. If we think of the isometries of a regular ''n''-gon: for odd ''n'' there are rotations in the group between every pair of mirrors, while for even ''n'' only half of the mirrors can be reached from one by these rotations. Geometrically, in an odd polygon every axis of symmetry passes through a vertex and a side, while in an even polygon there are two sets of axes, each corresponding to a conjugacy class: those that pass through two vertices and those that pass through two sides.

Algebraically, this is an instance of the conjugate [[Sylow theorem]] (for ''n'' odd): for ''n'' odd, each reflection, together with the identity, form a subgroup of order 2, which is a [[Sylow subgroup|Sylow 2-subgroup]] ({{nowrap|2 {{=}} 2{{sup|1}}}} is the maximum power of 2 dividing {{nowrap|2''n'' {{=}} 2[2''k'' + 1]}}), while for ''n'' even, these order 2 subgroups are not Sylow subgroups because 4 (a higher power of 2) divides the order of the group.

For ''n'' even there is instead an [[outer automorphism]] interchanging the two types of reflections (properly, a class of outer automorphisms, which are all conjugate by an inner automorphism).