# Difference between revisions of "Quaternion group"

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## Revision as of 20:07, 29 June 2011

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 holomorph of the ring .
- It is the holomorph of the cyclic group of order 4.
- 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

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

## Families

- The construction of the quaternion group can be mimicked for other primes giving, in general, a non-Abelian group of order . The general construction involves taking a semidirect product of the cyclic group of order with a subgroup of order in the automorphism group, say the subgroup generated by the automorphism taking an element to its .
- The quaternion group also generalizes to the family of dicyclic groups (also known as binary dihedral groups) and also to the family of generalized quaternion groups (which are the dicyclic groups whose order is a power of 2).
- The quaternion group is part of a larger family of -groups called extraspecial groups. An extraspecial group is a group of prime power order whose center, commutator subgroup and Frattini subgroup coincide, and are all cyclic of prime order.

## 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

### Arithmetic functions of an element-counting nature

`Further information: element structure of quaternion group`

### Arithmetic functions of a subgroup-counting nature

`Further information: subgroup structure of quaternion group`

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

number of subgroups | 6 | ||

number of conjugacy classes of subgroups | 6 | ||

number of normal subgroups | 6 | groups with same order and number of normal subgroups | groups with same number of normal subgroups | |

number of automorphism classes of subgroups | 4 |

### Lists of numerical invariants

List | Value | Explanation/comment |
---|---|---|

conjugacy class sizes | are each conjugacy classes of non-central elements. | |

degrees of irreducible representations | See linear representation theory of quaternion group | |

order statistics | ||

orders of subgroups | See subgroup structure of quaternion group |

## Group properties

Want to compare and contrast group properties with other groups of the same order? Check out groups of order 8#Group properties

## Subgroups

`Further information: Subgroup structure of quaternion group`

Automorphism class of subgroups | List of subgroups | Isomorphism class | Order of subgroups | Index of subgroups | Number of conjugacy classes (=1 iff automorph-conjugate subgroup) | Size of each conjugacy class (=1 iff normal subgroup) | Total number of subgroups (=1 iff characteristic subgroup) | Isomorphism class of quotient (if exists) | Nilpotency class |
---|---|---|---|---|---|---|---|---|---|

trivial subgroup | trivial subgroup | 1 | 8 | 1 | 1 | 1 | quaternion group | 0 | |

center of quaternion group | cyclic group:Z2 | 2 | 4 | 1 | 1 | 1 | Klein four-group | 1 | |

cyclic maximal subgroups of quaternion group | |
cyclic group:Z4 | 4 | 2 | 3 | 1 | 3 | cyclic group:Z2 | 1 |

whole group | quaternion group | 8 | 1 | 1 | 1 | 1 | trivial group | 2 | |

Total (4 rows) | -- | -- | -- | -- | 6 | -- | 6 | -- | -- |

## Subgroup-defining functions and associated quotient-defining functions

## Other associated constructs

Associated construct | Value (isomorphism class) | Comment |
---|---|---|

Automorphism group | symmetric group:S4 | |

Outer automorphism group | symmetric group:S3 | |

Inner holomorph | inner holomorph of D8 | The inner holomorphs of and the quaternion group are isomorphic. |

## Supergroups

`Further information: Supergroups of quaternion group`

## Implementation in GAP

### Group ID

This finite group has order 8 and has ID 4 among the groups of order 8 in GAP's SmallGroup library. For context, there are 5 groups of order 8. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(8,4)`

For instance, we can use the following assignment in GAP to create the group and name it :

`gap> G := SmallGroup(8,4);`

Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:

`IdGroup(G) = [8,4]`

or just do:

`IdGroup(G)`

to have GAP output the group ID, that we can then compare to what we want.

### Short descriptions

Description | Functions used | Mathematical comment |
---|---|---|

SylowSubgroup(SL(2,3),2) |
SylowSubgroup and SL | The -Sylow subgroup of special linear group:SL(2,3) |

ExtraspecialGroup(2^3,'-') |
ExtraspecialGroup | The extraspecial group of order and '-' type |

SylowSubgroup(SL(2,5),2) |
SylowSubgroup and SL | The -Sylow subgroup of special linear group:SL(2,5) |