# Difference between revisions of "Quaternion group"

(→Implementation in GAP) |
(→Arithmetic functions) |
||

Line 78: | Line 78: | ||

==Arithmetic functions== | ==Arithmetic functions== | ||

− | {{ | + | {{compare and contrast arithmetic functions|order = 8}} |

− | {| class=" | + | {| class="sortable" border="1" |

− | ! Function !! Value !! Explanation | + | ! Function !! Value !! Similar groups !! Explanation |

|- | |- | ||

− | | [[underlying prime of p-group]] || [[arithmetic function value::underlying prime of p-group;2|2]] || | + | | [[underlying prime of p-group]] || [[arithmetic function value::underlying prime of p-group;2|2]] || || |

|- | |- | ||

− | | | + | | {{arithmetic function value order|8}} || |

|- | |- | ||

− | | | + | | {{arithmetic function value order p-log|3}} || |

|- | |- | ||

− | | | + | | {{arithmetic function value given order|exponent of a group|4|8}} || Cyclic subgroup of order four. |

|- | |- | ||

− | | | + | | {{arithmetic function value given order and p-log|prime-base logarithm of exponent|2|8|3}} || |

|- | |- | ||

− | | | + | | {{arithmetic function value given order and p-log|nilpotency class|2|8|3}} || |

|- | |- | ||

− | | | + | | {{arithmetic function value given order and p-log|derived length|2|8|3}} || |

|- | |- | ||

− | | | + | | {{arithmetic function value given order and p-log|Frattini length|2|8|3}} || |

|- | |- | ||

− | | | + | | {{arithmetic function value given order and p-log|minimum size of generating set|2|8|3}} || Generators of two cyclic subgroups of order four. |

|- | |- | ||

− | | | + | | {{arithmetic function value given order and p-log|subgroup rank of a group|2|8|3}} || All proper subgroups are cyclic. |

|- | |- | ||

− | | | + | | {{arithmetic function value given order and p-log|rank of a p-group|1|8|3}} || All abelian subgroups are cyclic. |

|- | |- | ||

− | | | + | | {{arithmetic function value given order and p-log|normal rank of a p-group|1|8|3}} || All abelian normal subgroups are cyclic. |

|- | |- | ||

− | | | + | | {{arithmetic function value given order and p-log|characteristic rank of a p-group|1|8|3}} || All abelian characteristic subgroups are cyclic. |

− | |||

− | | | ||

− | | | ||

− | | | ||

|} | |} | ||

## Revision as of 21:44, 10 July 2010

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]

## 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.

### 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

### Upto conjugacy

The quaternion group has five conjugacy classes:

- The identity element: This has order 1 and size 1
- The element : This has order 2 and size 1
- The two-element conjugacy class comprising : This has order 4 and size 2
- The two-element conjugacy class comprising : This has order 4 and size 2
- The two-element conjugacy class comprising : This has order 4 and size 2

### Upto automorphism

Under the action of automorphisms, the last three conjugacy classes get merged, so there are three equivalence classes, of sizes 1, 1, and 6.

## 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

### 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 with other groups of the same order? Check out groups of order 8#Group properties.

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

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

nilpotent group | Yes | Prime power order implies nilpotent | Smallest nilpotent non-abelian group, along with dihedral group:D8. |

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

supersolvable group | Yes | Metacyclic implies supersolvable | |

solvable group | Yes | Metacyclic implies solvable | |

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

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

monolithic group | Yes | Unique minimal normal subgroup of order two | |

one-headed group | No | Three distinct maximal normal subgroups of order four | |

SC-group | No | ||

ACIC-group | Yes | Every automorph-conjugate subgroup is characteristic | |

ambivalent group | Yes | ||

rational group | Yes | Any two elements that generate the same cyclic group are conjugate | Thus, all characters are integer-valued. |

rational-representation group | Yes | A two-dimensional representation that is not rational. | Contrast with dihedral group:D8, that is rational-representation. |

maximal class group | Yes | ||

group of nilpotency class two | Yes | ||

extraspecial group | Yes | ||

special group | Yes | ||

Frattini-in-center group | Yes | ||

Frobenius group | No | Frobenius groups are centerless, and this group isn't. | |

Camina group | Yes | extraspecial implies Camina | |

group in which every element is automorphic to its inverse | Yes | Follows from being an ambivalent group | |

group in which any two elements generating the same cyclic subgroup are automorphic | Yes | Follows from being a rational group | |

group in which every element is order-automorphic | Yes | ||

directly indecomposable group | Yes | ||

centrally indecomposable group | Yes | ||

splitting-simple group | Yes |

## Subgroups

`Further information: Subgroup structure of quaternion group`

The quaternion group has six subgroups:

- The trivial subgroup (1)
- The center, which is the unique minimal subgroup. This is a two-element subgroup comprising (1)
- The three cyclic subgroups of order four, generated by respectively. These are all normal, but are automorphs of each other (3)
- The whole group (1)

### Normal subgroups

All subgroups are normal. The subgroups are the whole group, the trivial subgroup, the center, and three copies of the cyclic group on 4 elements. This makes the quaternion group a Dedekind group.

### Characteristic subgroups

There are only three characteristic subgroups: the whole group, the trivial subgroup and the center.

## Subgroup-defining functions

Subgroup-defining function | Subgroup type in list | Page on subgroup embedding | Isomorphism class | Comment |
---|---|---|---|---|

Center | (2) | Center of quaternion group | Cyclic group:Z2 | Prime power order implies not centerless |

Commutator subgroup | (2) | Center of quaternion group | Cyclic group:Z2 | |

Frattini subgroup | (2) | Center of quaternion group | Cyclic group:Z2 | The three maximal subgroups of order four intersect here. |

Socle | (2) | Center of quaternion group | Cyclic group:Z2 | This subgroup is the unique minimal normal subgroup, i.e.,the monolith, and the group is monolithic. Also, minimal normal implies central in nilpotent. |

## Quotient-defining functions

Quotient-defining function | Isomorphism class | Comment |
---|---|---|

Inner automorphism group | Klein four-group | It is the quotient by the center, which is of order two. |

Abelianization | Klein four-group | It is the quotient by the commutator subgroup, which is cyclic of order two. |

Frattini quotient | Klein four-group | It is the quotient by the Frattini subgroup, which is cyclic of order two. |

## 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) |