# Quaternion group

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## Contents |

## 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.

Confused about presentations in general or this one in particular? If you're new to this stuff, check out constructing quaternion group from its presentation. Sophisticated group theorists can read equivalence of presentations of dicyclic group

### 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

### 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

### Important properties

Property | Satisfied | Explanation | Comment |
---|---|---|---|

group of prime power order | Yes | ||

nilpotent group | Yes | prime power order implies nilpotent | |

supersolvable group | Yes | via nilpotent: finite nilpotent implies supersolvable | |

solvable group | Yes | via nilpotent: nilpotent implies solvable | |

abelian group | No | and don't commute | Smallest non-abelian group of prime power order |

metacyclic group | Yes | Cyclic normal subgroup of order four, cyclic quotient of order two | |

Dedekind group | Yes | Every subgroup is normal | Smallest non-abelian Dedekind group |

T-group | Yes | Dedekind implies T-group |

### Other 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

## Automorphisms and endomorphisms

`Further information: endomorphism structure of quaternion group`

Construct | Value | Order | Second part of GAP ID (if group) |
---|---|---|---|

endomorphism monoid | ? | ? | -- |

automorphism group | symmetric group:S4 | 24 | 12 |

inner automorphism group | Klein four-group | 4 | 2 |

outer automorphism group | symmetric group:S3 | 6 | 1 |

group of class-preserving automorphisms | Klein four-group | 4 | 2 |

group of IA-automorphisms | Klein four-group | 4 | 2 |

quotient of class-preserving automorphism group by inner automorphism group | trivial group | 1 | 1 |

quotient of IA-automorphism group by inner automorphism group | trivial group | 1 | 1 |

group of center-fixing automorphisms | symmetric group:S4 | 24 | 12 |

extended automorphism group | direct product of S4 and Z2 | 48 | 48 |

holomorph | ? | 192 | |

inner holomorph | inner holomorph of D8 ( and the quaternion group have the same holomorph) | 32 | 49 |

## Linear representation theory

`Further information: linear representation theory of quaternion group`

### Summary

The **quaternion group** is one of the few examples of a rational group that is not a rational-representation group. In other words, all its characters over the complex numbers are rational-valued, but not every representation of it can be realized over the rationals.

The character table of the quaternion group is the same as that of the dihedral group of order eight. Note, however, that the fields of realization for the representations differ, because one of the representations of the quaternion group has Schur index two.

Item | Value |
---|---|

Degrees of irreducible representations over a splitting field (such as or ) | 1,1,1,1,2 maximum: 2, lcm: 2, number: 5, sum of squares: 8 |

Schur index values of irreducible representations | 1,1,1,1,2 (characteristic zero) maximum: 2, lcm: 2 1,1,1,1,1 (characteristic other than 0,2) |

Smallest ring of realization for all irreducible representations (characteristic zero) | There are multiple candidates. where is a square root of , equivalently , the ring of Gaussian integers is one candidate. Another is or . More generally, any ring of the form where is a ring of realization for all irreducible representations. In particular, works for any rational . |

Minimal splitting field (i.e., field of realization) for all irreducible representations (characteristic zero) | There are multiple candidates. or works, so does or . More generally, where is a splitting field. In particular, works for any rational . See minimal splitting field need not be unique, minimal splitting field need not be cyclotomic |

Ring generated by character values (characteristic zero) | |

Field generated by character values (characteristic zero) | (hence it is a rational group) See also: Field generated by character values need not be a splitting field|rational not implies rational-representation |

Condition for being a splitting field for this group | Sufficient condition: the characteristic is not two and there exist in the field such that . In particular, every finite field of characteristic not two is a splitting field, because every element of a finite field is expressible as a sum of two squares and in particular, is a sum of two squares in any finite field. |

Minimal splitting field (characteristic ) | The prime field |

Smallest size splitting field | field:F3, i.e., the field of three elements. |

Orbit structure of irreducible representations over splitting field under automorphism group | orbit sizes: 1 (degree 1 representation), 3 (degree 1 representations), 1 (degree 2 representation) number: 3 |

Orbit structure of irreducible representations over splitting field under multiplicative action of one-dimensional representations, i.e., up to projective equivalence | orbit sizes: 4 (degree 1 representations), 1 (degree 2 representation) number: 2 |

Degrees of irreducible representations over a non-splitting field, e.g., the field of rational numbers or the field of real numbers | 1,1,1,1,4 number: 5 |

Groups with same character table | Dihedral group:D8 |

### Character table

This character table works over characteristic zero and over any other characteristic not equal to two once we reduce the entries mod the characteristic:

Representation/Conjugacy class | (identity; size 1) | (size 1) | (size 2) | (size 2) | (size 2) |
---|---|---|---|---|---|

Trivial representation | 1 | 1 | 1 | 1 | 1 |

-kernel | 1 | 1 | 1 | -1 | -1 |

-kernel | 1 | 1 | -1 | 1 | -1 |

-kernel | 1 | 1 | -1 | -1 | 1 |

2-dimensional | 2 | -2 | 0 | 0 | 0 |

## Distinguishing features

### Smallest of its kind

- This is a non-abelian nilpotent group of smallest possible order, along with dihedral group:D8.
- This is a non-abelian Dedekind group (or Hamiltonian group) of smallest possible order.
**Dedekind**means that every subgroup is normal.

### Different from others of the same order

- It is the only non-abelian Dedekind group of its order.
- It is the only non-abelian T-group of its order.
- It is the only group of its order for which the rank (in the sense of the maximum possible rank of an abelian subgroup) is
*strictly*smaller than the minimum size of generating set: For this group, the former is 1 and the latter is 2.

## GAP implementation

### 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.

### Hall-Senior number

This group of prime power order has order 8 and has Hall-Senior number 5 among the groups of order 8. This information can be used to construct the group in GAP using the Gap3CatalogueGroup function as follows:

`Gap3CatalogueGroup(8,5)`

WARNING: There is some disagreement between the GAP 3 catalogue numbers and the Hall-Senior numbers for some abelian groups, but it does not affect this group.

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

`gap> G := Gap3CatalogueGroup(8,5);`

Conversely, to check whether a given group is in fact the group we want, we can use GAP's Gap3CatalogueIdGroup function:

`Gap3CatalogueIdGroup(G) = [8,5]`

or just do:

`Gap3CatalogueIdGroup(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) |