# Orthogonal group:O(2,R)

From Groupprops

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this groupView a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## Definition

### Main definition

This group is defined as the group of matrices with real entries such that is the identity matrix. Equivalently, it can be defined as:

.

In fact, there are only two possible forms of such matrices:

.

The subgroup of matrices with determinant (i.e., the matrices with ) is the special orthogonal group . It has index two and is isomorphic to the circle group.

This group is a particular case of an orthogonal group over reals and hence of an orthogonal group.

### Alternative definitions

This group can be defined in the following other ways:

- It is the generalized dihedral group corresponding to the circle group, i.e., it is the semidirect product of the circle group by a group of order two acting by the inverse map.