# Quaternion group

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

### Definition by presentation

The quaternion group has the following presentation:

The identity is denoted , the common element is denoted , and the elements are denoted respectively.

### Verbal definitions

The **quaternion group** is a group with eight elements, which can be described in any of the following ways:

- It is the group comprising eight elements where 1 is the identity element, and all the other elements are squareroots of , such that and further, (the remaining relations can be deduced from these).
- It is the dicyclic group with parameter 2, viz .
- It is the Fibonacci group .

### Multiplication table

In the table below, the row element is multiplied on the left and the column element on the right.

Element | ||||||||
---|---|---|---|---|---|---|---|---|

## Position in classifications

Type of classification | Name in that classification |
---|---|

GAP ID | (8,4), i.e., the 4th among the groups of order 8 |

Hall-Senior number | 5 among groups of order 8 |

Hall-Senior symbol |

## Elements

`Further information: Element structure of quaternion group`

### Conjugacy class structure

Conjugacy class | Size of conjugacy class | Order of elements in conjugacy class | Centralizer of first element of class |
---|---|---|---|

1 | 1 | whole group | |

1 | 2 | whole group | |

2 | 4 | , same as | |

2 | 4 | -- same as | |

2 | 4 | -- same as |

### Automorphism class structure

Equivalence class (orbit) under action of automorphisms | Size of equivalence class (orbit) | Number of conjugacy classes in it | Size of each conjugacy class | Order of elements |
---|---|---|---|---|

1 | 1 | 1 | 1 | |

1 | 1 | 1 | 2 | |

6 | 3 | 2 | 4 |

## Arithmetic functions

### Basic arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 8#Arithmetic functions