# Special orthogonal group for the standard dot product

From Groupprops

This article defines a natural number-parametrized system of algebraic matrix groups. In other words, for every field and every natural number, we get a matrix group defined by a system of algebraic equations. The definition may also generalize to arbitrary commutative unital rings, though the default usage of the term is over fields.

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## Contents

## Definition

### Definition with symbols

Let be a natural number and a field. Then the special orthogonal group of order over the field , denoted , is defined as the group of all matrices such that and .

A group is termed a **special orthogonal group** if it occurs as for some natural number and field .

## Relation with other linear algebraic groups

### Supergroups

- Orthogonal group
- Special affine orthogonal group
- Affine orthogonal group
- Special orthogonal similitude group
- Orthogonal similitude group

## Particular cases

### Finite fields

Size of field | Order of matrices | Common name for the special orthogonal group | The special orthogonal group as embedded in the special linear group |
---|---|---|---|

1 | Trivial group | Trivial subgroup of trivial group | |

2 | Elementary abelian group of order | ||

2 | 2 | Cyclic group:Z2 | Two-element subgroups of symmetric group:S3 |

3 | 2 | Cyclic group:Z4 | Cyclic four-subgroups of special linear group:SL(2,3) |

4 | 2 | Klein four-group | Klein four-subgroups of alternating group:A5 |

5 | 2 | Cyclic group:Z4 | Cyclic four-subgroups of special linear group:SL(2,5) |

7 | 2 | Cyclic group:Z8 | |

8 | 2 | Elementary abelian group of order eight | |

9 | 2 | Cyclic group:Z8 | |

17 | 2 | Cyclic group:Z16 | |

2 | 3 | Cyclic group:Z6 | |

3 | 3 | Symmetric group:S4 | |

4 | 3 | Alternating group:A5 | |

5 | 3 | Symmetric group:S5 |