# Group of unit quaternions

From Groupprops

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## Contents

## Definition

This group is denoted and is defined in a number of equivalent ways.

### As the group of unit quaternions

Denote by the division ring of Hamiltonian quaternions. The group we are interested in is the multiplicative subgroup of comprising those unit quaternions satisfying . Note that (and are allowed to be equal). Explicitly, the multiplication is given by:

The identity element is:

The inverse is given by:

### As the special unitary group

The group can also be defined as the special unitary group of degree two over the field of complex numbers. It is denoted or .

### Structures

The group has the following structures:

- It is a real Lie group (note that it is
*not*a complex Lie group). - It is a linear algebraic group over the field of real numbers (note that it is not algebraic over the complex numbers).
- It is a topological group.

## Arithmetic functions

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

dimension of an algebraic group | 3 | As : As : (note that is a group only for ) | |

dimension of a real Lie group | 3 | As : As : (note that is a group only for ) |

## Group properties

### Abstract group properties

Property | Satisfied? | Explanation |
---|---|---|

abelian group | No | Follows from center of general linear group is group of scalar matrices over center |

nilpotent group | No | Follows from special linear group is quasisimple |

solvable group | No | Follows from special linear group is quasisimple |

simple group | No | Has a proper nontrivial center, also has normal subgroup . |

almost simple group | No | Has a nontrivial center. |

quasisimple group | Yes | It is a perfect group and its quotient by its center is SO(3,R) which is simple. |

almost quasisimple group | Yes | Follows from being quasisimple. |

divisible group | Yes | For any element of the group and any natural number , the element has a root in the group. |

### Topological/Lie group properties

The topology here is the subspace topology from the Euclidean topology on the set of all matrices, which is identified with the Euclidean space .

Property | Satisfied? | Explanation |
---|---|---|

connected topological group | Yes | easy to see from geometric description as . |

compact group | Yes | Easy to see from geometric description as . |