# Special orthogonal group over reals

From Groupprops

## Contents

## Definition

For any natural number , the **special orthogonal group over reals** of degree , denoted or , is defined as the following group:

This can also be described as the group of linear transformations of that are orientation-preserving and also preserve the dot product.

## Viewpoints

### As an algebraic group

### As a Lie group

### As a topological group

## Particular cases

Value of | Name of group | Special comments |
---|---|---|

1 | trivial group | none |

2 | circle group | the only nontrivial abelian case |

3 | special orthogonal group:SO(3,R) | the smallest non-abelian case. Also, has (group of unit quaternions) as a double cover |

4 | special orthogonal group:SO(4,R) |