View source for Quaternion group

{{particular group}}
{{TOCright}}
[[importance rank::2| ]]
==Definition==

===Definition by presentation===

The quaternion group has the following [[presentation of a group|presentation]]:

<math>\langle i,j,k \mid i^2 = j^2 = k^2 = ijk \rangle</math>

The identity is denoted <math>1</math>, the common element <math>i^2 = j^2 = k^2 = ijk</math> is denoted <math>-1</math>, and the elements <math>i^3, j^3, k^3</math> are denoted <math>-i,-j,-k</math> respectively.

{{quotation|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 <math>1,-1,i,-i,j,-j,k,-k</math> where 1 is the identity element, <math>(-1)^2 = 1</math> and all the other elements are squareroots of <math>-1</math>, such that <math>(-1)i = -i, (-1)j = -j, (-1)k= -k</math> and further, <math>ij = k, ji = -k, jk = i, kj = -i, ki = j, ik = -j</math> (the remaining relations can be deduced from these).
* It is the {{dicyclic group}} with parameter 2, viz <math>Dic_2</math>.
* It is the [[member of family::Fibonacci group]] <math>F(2,3)</math>.

===Multiplication table===

{{#lst:element structure of quaternion group|multiplication table}}

==Position in classifications==

{| class="sortable" border="1"
! 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 || <math>8\Gamma_2a_2</math>
|}

==Elements==

{{further|[[Element structure of quaternion group]]}}

===Conjugacy class structure===
{{#lst:element structure of quaternion group|conjugacy and automorphism class structure}}

==Arithmetic functions==

===Basic arithmetic functions===

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

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation
|-
| [[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.
|}

===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}}

===Important properties===

{| 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 || 
|}

===Other properties===

{| class="sortable" border="1"
!Property !! Satisfied !! Explanation !! Comment
|-
|[[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]] || No || [[Faithful irreducible representation of quaternion group|A two-dimensional representation that is not rational]]. || Contrast with [[dihedral group:D8]], that is rational-representation. See also [[linear representation theory of dihedral group:D8]] and [[linear representation theory of quaternion group]].
|-
| [[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 || ||
|-
|[[Satisfies property::Schur-trivial group]] || Yes || See [[group cohomology of quaternion group]] ||
|}

==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}}

==Linear representation theory==

{{further|[[linear representation theory of quaternion group]]}}

===Summary===

{{#lst:linear representation theory of quaternion group|summary}}

===Character table===

{{#lst:linear representation theory of quaternion group|character table}}

==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}}

{{HallSenior|8|5}}

===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)]]
|}