(Note that some authors use the notation D_{n} instead of Wikipedia's notation D_{2n}.)
The simplest dihedral group is D_{4}, which is generated by a rotation r of 180 degrees, and a reflection f across the yaxis. The elements of D_{4} can then be represented as {e, r, f, rf}, where e is the identity or null transformation.
D_{4} is isomorphic to the Klein fourgroup.
If the order of D_{2n} is greater than 4, the operations of rotation and reflection in general do not commute and D_{2n} is not abelian; for example, in D_{8}, a rotation of 90 degrees followed by a reflection yields a different result from a reflection followed by a rotation of 90 degrees.
Whatever the order of the dihedral group, the rotation r and the reflection f always satisfy
Thus, beyond their obvious application to problems of symmetry in the plane, these groups are among the simplest examples of nonabelian groups, and as such arise frequently as easy counterexamples to theorems which are restricted to abelian groups.
Some equivalent definitions of D_{2n} are:
The number of subgroups of D_{2n} (n ≥ 3), is equal to d(n) + σ(n), where d(n) is the number of positive divisors of n and σ(n) is the sum of the positive divisors of n.
In addition to the finite dihedral groups, there is the infinite dihedral group D_{∞}. Every dihedral group is generated by a rotation r and a reflection; if the rotation is a rational multiple of a full rotation, then there is some integer n such that r^{n} will be the identity. If the rotation is not a rational multiple, then there is no such n; the resulting group is then called D_{∞}. It has presentation ({a,b}; {a², b²}}, and is isomorphic to a semidirect product of Z and C_{2}.
D_{∞} can also be visualized as the automorphism group of the graph consisting of a path infinite to both sides.
Finally, if H is any nontrivial finite abelian group, we can speak of the generalized dihedral group of H (sometimes written Dih(H)). This group is a semidirect product of H and C_{2}, with order 2*order(H), a normal subgroup of index 2 isomorphic to H, and having an element f such that, for all x in H, f^{ 1} x f = x^{ 1}.
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