# Dihedral Group

In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. It is well-known and quite trivial to prove that a group generated by two involutions is a dihedral group.

### Other articles related to "dihedral group, dihedral groups, groups, group":

Quasidihedral Group
... In mathematics, the quasi-dihedral groups and semi-dihedral groups are non-abelian groups of order a power of 2 ... equal to 4, there are exactly four isomorphism classes of nonabelian groups of order 2n which have a cyclic subgroup of index 2 ... Two are well established, the generalized quaternion group and the dihedral group, but the other two have conflicting names, so this article discusses both groups ...
Nielsen Transformation - Examples
... The dihedral group of order 10 has two Nielsen equivalence classes of generating sets of size 2 ... A very important generating set of a dihedral group is the generating set from its presentation as a Coxeter group ... Such a generating set for a dihedral group of order 10 consists of any pair of elements of order 2, such as ...
Dihedral Group - Generalizations
... There are several important generalizations of the dihedral groups The infinite dihedral group is an infinite group with algebraic structure similar to the finite dihedral groups ... It can be viewed as the group of symmetries of the integers ... The orthogonal group O(2), i.e ...
Infinite Dihedral Group - Definition
... 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 rn is ... is not a rational multiple of a full rotation, then there is no such n and the resulting group has infinitely many elements and is called Dih∞ ... It is the automorphism group of the graph consisting of a path infinite to both sides ...

### Famous quotes containing the word group:

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