Editing Dihedral group (section)

==Properties==

The properties of the dihedral groups {{math|D{{sub|''n''}}}} with {{math|''n'' ≥ 3}} depend on whether {{math|''n''}} is even or odd. For example, the [[center of a group|center]] of {{math|D{{sub|''n''}}}} consists only of the identity if ''n'' is odd, but if ''n'' is even the center has two elements, namely the identity and the element r<sup>''n''/2</sup> (with D<sub>''n''</sub> as a subgroup of O(2), this is [[Inversion (discrete mathematics)|inversion]]; since it is [[scalar multiplication]] by &minus;1, it is clear that it commutes with any linear transformation).

In the case of 2D isometries, this corresponds to adding inversion, giving rotations and mirrors in between the existing ones.

For ''n'' twice an odd number, the abstract group {{math|D{{sub|''n''}}}} is isomorphic with the [[direct product of groups|direct product]] of {{math|D{{sub|''n'' / 2}}}} and {{math|Z{{sub|2}}}}.
Generally, if ''m'' [[divisor|divides]] ''n'', then {{math|D{{sub|''n''}}}} has ''n''/''m'' [[subgroup]]s of type {{math|D{{sub|''m''}}}}, and one subgroup <math>\mathbb{Z}</math><sub>''m''</sub>. Therefore, the total number of subgroups of {{math|D{{sub|''n''}}}} (''n''&nbsp;≥&nbsp;1), is equal to ''d''(''n'')&nbsp;+&nbsp;σ(''n''), where ''d''(''n'') is the number of positive [[divisor]]s of ''n'' and ''σ''(''n'') is the sum of the positive divisors of&nbsp;''n''.  See [[list of small groups]] for the cases&nbsp;''n''&nbsp;≤&nbsp;8.

The dihedral group of order 8 (D<sub>4</sub>) is the smallest example of a group that is not a [[T-group (mathematics)|T-group]]. Any of its two [[Klein four-group]] subgroups (which are normal in D<sub>4</sub>) has as normal subgroup order-2 subgroups generated by a reflection (flip) in D<sub>4</sub>, but these subgroups are not normal in D<sub>4</sub>.

===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).