Quaternion group: Difference between revisions

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| {{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|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.
| {{arithmetic function value given order and p-log|characteri
|}
 
===Arithmetic functions of an element-counting nature===
 
{{further|[[element structure of quaternion group]]}}
 
{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation
|-
| {{arithmetic function value given order|number of conjugacy classes|5|8}} || See [[element structure of dicyclic groups]].
|-
| {{arithmetic function value given order|number of equivalence classes under real conjugacy|5|8}} || Same as number of conjugacy classes, because the group is an [[ambivalent group]].
|-
| {{arithmetic function value given order|number of conjugacy classes of real elements|5|8}} || Same as number of conjugacy clases, because the group is an [[ambivalent group]].
|-
| {{arithmetic function value given order|number of equivalence classes under rational conjugacy|5|8}} || Same as number of conjugacy classes, because the group is a [[rational group]] (though not a [[rational representation group]]).
|-
| {{arithmetic function value given order|number of conjugacy classes of rational elements|5|8}} || Same as number of conjugacy classes, because the group is a [[rational group]] (though not a [[rational representation group]]).
|}
===Arithmetic functions of a subgroup-counting nature===
 
{{further|[[subgroup structure of quaternion group]]}}
 
{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation
|-
| {{arithmetic function value|number of subgroups|6}} || ||
|-
| {{arithmetic function value|number of conjugacy classes of subgroups|6}} || ||
|-
| {{arithmetic function value given order|number of normal subgroups|6|8}} ||
|-
| {{arithmetic function value|number of automorphism classes of subgroups|4}} || ||
|}
 
===Lists of numerical invariants===
 
{| class="sortable" border="1"
! List !! Value !! Explanation/comment
|-
| [[conjugacy class size set|conjugacy class sizes]] || <math>1,1,2,2,2</math> || <math>\pm i, \pm j, \pm k</math> are each conjugacy classes of non-central elements.
|-
| [[degrees of irreducible representations]] || <math>1,1,1,1,2</math> || See [[linear representation theory of quaternion group]]
|-
| [[order statistics of a finite group|order statistics]] || <math>1 \mapsto 1, 2 \mapsto 1, 4 \mapsto 6</math> ||
|-
| orders of subgroups || <math>1,2,4,4,4,8</math> || See [[subgroup structure of quaternion group]]
|}
 
==Group properties==
 
{{compare and contrast group properties|order = 8}}
 
{| class="sortable" border="1"
!Property !! Satisfied !! Explanation !! Comment
|-
| {{group properties because p-group}}
|-
|[[Dissatisfies property::abelian group]] || No || <math>i</math> and <math>j</math> don't commute || Smallest non-abelian [[satisfies property::group of prime power order]]
|-
|[[Satisfies property::metacyclic group]] || Yes || Cyclic normal subgroup of order four, cyclic quotient of order two ||
|-
|[[Satisfies property::Dedekind group]] || Yes|| Every subgroup is normal || Smallest non-abelian Dedekind group
|-
|[[Satisfies property::T-group]] || Yes || Dedekind implies T-group ||
|-
|[[Satisfies property::monolithic group]] || Yes|| Unique minimal normal subgroup of order two ||
|-
|[[Dissatisfies property::one-headed group]] || No || Three distinct maximal normal subgroups of order four ||
|-
|[[Dissatisfies property::SC-group]] || No ||  ||
|-
|[[Satisfies property::ACIC-group]] || Yes || Every [[automorph-conjugate subgroup]] is [[characteristic subgroup|characteristic]] ||
|-
| [[Satisfies property::ambivalent group]] || Yes || ||
|-
|[[Satisfies property::rational group]] || Yes || Any two elements that generate the same cyclic group are conjugate || Thus, all characters are integer-valued.
|-
|[[Dissatisfies property::rational-representation group]] || Yes || A two-dimensional representation that is not rational. || Contrast with [[dihedral group:D8]], that is rational-representation.
|-
| [[Satisfies property::maximal class group]] || Yes || ||
|-
| [[Satisfies property::group of nilpotency class two]] || Yes|| ||
|-
| [[Satisfies property::extraspecial group]] || Yes || ||
|-
| [[Satisfies property::special group]] || Yes || ||
|-
| [[Satisfies property::Frattini-in-center group]] || Yes || ||
|-
|[[Dissatisfies property::Frobenius group]] || No || Frobenius groups are centerless, and this group isn't. ||
|-
|[[Satisfies property::Camina group]] || Yes || [[extraspecial implies Camina]] ||
|-
|[[Satisfies property::group in which every element is automorphic to its inverse]] || Yes || Follows from being an [[ambivalent group]] ||
|-
|[[Satisfies property::group in which any two elements generating the same cyclic subgroup are automorphic]] || Yes || Follows from being a [[rational group]] ||
|-
|[[Satisfies property::group in which every element is order-automorphic]] || Yes || ||
|-
|[[Satisfies property::directly indecomposable group]] || Yes || ||
|-
|[[Satisfies property::centrally indecomposable group]] || Yes || ||
|-
|[[Satisfies property::splitting-simple group]] || Yes || ||
|}
 
==Subgroups==
{{further|[[Subgroup structure of quaternion group]]}}
[[Image:Q8latticeofsubgroups.png|500px]]
{{#lst:subgroup structure of quaternion group|summary}}
 
==Subgroup-defining functions and associated quotient-defining functions==
 
{{#lst:subgroup structure of quaternion group|sdf summary}}
==Automorphisms and endomorphisms==
 
{{further|[[endomorphism structure of quaternion group]]}}
 
{{#lst:endomorphism structure of quaternion group|summary}}
 
==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 of a p-group|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==
 
{{GAP ID|8|4}}
 
===Short descriptions===
 
{| class="sortable" border="1"
! Description !! Functions used !! Mathematical comment
|-
| <tt>SylowSubgroup(SL(2,3),2)</tt> || [[GAP:SylowSubgroup|SylowSubgroup]] and [[GAP:SL|SL]] || The <math>2</math>-Sylow subgroup of [[special linear group:SL(2,3)]]
|-
| <tt>ExtraspecialGroup(2^3,'-')</tt> || [[GAP:ExtraspecialGroup|ExtraspecialGroup]] || The extraspecial group of order <math>2^3</math> and '-' type
|-
| <tt>SylowSubgroup(SL(2,5),2)</tt> || [[GAP:SylowSubgroup|SylowSubgroup]] and [[GAP:SL|SL]] || The <math>2</math>-Sylow subgroup of [[special linear group:SL(2,5)]]
|}

Revision as of 12:01, 1 July 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 group
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Definition

Definition by presentation

The quaternion group has the following presentation:

⟨i,j,k∣i2=j2=k2=ijk⟩

The identity is denoted 1, the common element i2=j2=k2=ijk is denoted −1, and the elements i3,j3,k3 are denoted −i,−j,−k 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 1,−1,i,−i,j,−j,k,−k where 1 is the identity element, (−1)2=1 and all the other elements are squareroots of −1, such that (−1)i=−i,(−1)j=−j,(−1)k=−k and further, ij=k,ji=−k,jk=i,kj=−i,ki=j,ik=−j (the remaining relations can be deduced from these).
  • It is the dicyclic group with parameter 2, viz Dic2.
  • It is the Fibonacci group F(2,3).

Multiplication table

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

Element 1 −1 i −i j −j k −k
1 1 −1 i −i j −j k −k
−1 −1 1 −i i −j j −k k
i i −i −1 1 k −k −j j
−i −i i 1 −1 −k k j −j
j j −j −k k −1 1 i −i
−j −j j k −k 1 −1 −i i
k k −k j −j −i i −1 1
−k −k k −j j i −i 1 −1

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 8Γ2a2

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 1 whole group
{−1} 1 2 whole group
{i,−i} 2 4 {1,−1,i,−i}, same as ⟨i⟩
{j,−j} 2 4 {1,−1,j,−j} -- same as ⟨j⟩
{k,−k} 2 4 {1,−1,k,−k} -- same as ⟨k⟩

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 1 1 2
{i,−i,j,−j,k,−k} 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

Function Value Similar groups Explanation
underlying prime of p-group 2
order (number of elements, equivalently, cardinality or size of underlying set) 8 groups with same order
prime-base logarithm of order 3 groups with same prime-base logarithm of order
exponent of a group 4 groups with same order and exponent of a group | groups with same exponent of a group Cyclic subgroup of order four.
prime-base logarithm of exponent 2 groups with same order and prime-base logarithm of exponent | groups with same prime-base logarithm of order and prime-base logarithm of exponent | groups with same prime-base logarithm of exponent
nilpotency class 2 groups with same order and nilpotency class | groups with same prime-base logarithm of order and nilpotency class | groups with same nilpotency class
derived length 2 groups with same order and derived length | groups with same prime-base logarithm of order and derived length | groups with same derived length
Frattini length 2 groups with same order and Frattini length | groups with same prime-base logarithm of order and Frattini length | groups with same Frattini length
minimum size of generating set 2 groups with same order and minimum size of generating set | groups with same prime-base logarithm of order and minimum size of generating set | groups with same minimum size of generating set Generators of two cyclic subgroups of order four.
subgroup rank of a group 2 groups with same order and subgroup rank of a group | groups with same prime-base logarithm of order and subgroup rank of a group | groups with same subgroup rank of a group All proper subgroups are cyclic.
rank of a p-group 1 groups with same order and rank of a p-group | groups with same prime-base logarithm of order and rank of a p-group | groups with same rank of a p-group All abelian subgroups are cyclic.
normal rank of a p-group 1 groups with same order and normal rank of a p-group | groups with same prime-base logarithm of order and normal rank of a p-group | groups with same normal rank of a p-group All abelian normal subgroups are cyclic.
characteri
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