# Split special orthogonal group of degree two

From Groupprops

## Contents

## Definition

Suppose is a field. The **split special orthogonal group of degree two** over is defined as the subgroup of the general linear group of degree two over given as follows:

For characteristic not equal to two, an alternative definition, which gives a conjugate subgroup and hence an isomorphic group, is:

### Over a finite field

For a finite field , this group is denoted and is termed the orthogonal group of "+" type. It is also denoted where is the size of the field.

It turns out that:

- If is a power of 2, i.e., if is even, then the group is a dihedral group of order and degree .
- If is odd, then the group is a cyclic group of order .

## Arithmetic functions

### Over a finite field

We consider here the group where is a field (unique up to isomorphism) of size .

Function | Value | Similar groups | Explanation |
---|---|---|---|

order | Case odd: Case even: |

## Particular cases

(field size) | (underlying prime, field characteristic) | exponent on giving | special orthogonal group | order of group | second part of GAP ID (GAP ID is (order,2nd part) | Comments |
---|---|---|---|---|---|---|

2 | 2 | 1 | cyclic group:Z2 | 2 | 1 | |

3 | 3 | 1 | cyclic group:Z2 | 2 | 1 | |

4 | 2 | 2 | symmetric group:S3 | 6 | 1 | |

5 | 5 | 1 | cyclic group:Z4 | 4 | 1 | |

7 | 7 | 1 | cyclic group:Z6 | 6 | 2 | |

8 | 2 | 3 | dihedral group:D14 | 14 | 1 | |

9 | 3 | 2 | cyclic group:Z8 | 8 | 1 | |

11 | 11 | 1 | cyclic group:Z10 | 10 | 2 | |

13 | 13 | 1 | cyclic group:Z12 | 12 | 2 | |

16 | 2 | 4 | dihedral group:D30 | 30 | 3 | |

17 | 17 | 1 | cyclic group:Z16 | 16 | 1 |