View source for Dihedral group:D8

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

===Definition by presentation===

The '''dihedral group''' <math>D_8</math>, sometimes called <math>D_4</math>, also called the {{dihedral group}} of order eight or ''the dihedral group of degree four'' (since its natural action is on four elements), or sometimes the ''octic group'', is defined by the following [[presentation of a group|presentation]], with <math>e</math> denoting the identity element:

<math>\langle x,a \mid a^4 = x^2 = e, xax^{-1} = a^{-1}\rangle</math>

Here, the element <math>a</math> is termed the ''rotation'' or the ''generator of the cyclic piece'' and <math>x</math> is termed the ''reflection''.

{{quotation|Confused about presentations in general or this one in particular? If you're new to this stuff, check out [[constructing dihedral group:D8 from its presentation]]. Sophisticated group theorists need simply recall that [[presentation of semidirect product is disjoint union of presentations plus action by conjugation relations]]}}

===Geometric definition===

The '''dihedral group''' <math>D_8</math> (also called <math>D_4</math>) is defined as the group of all symmetries of the square (the regular 4-gon). This has a cyclic subgroup comprising rotations (which is the cyclic subgroup generated by <math>a</math>) and has four ''reflections'' each being an [[involution]]: reflections about lines joining midpoints of opposite sides, and reflections about diagonals.

===Definition as a permutation group===

{{further|[[D8 in S4]]}}

The group is (up to isomorphism) the subgroup of the symmetric group on <math>\{ 1,2,3,4 \}</math> given by:

<math>\! \{ (), (1,2,3,4), (1,3)(2,4), (1,4,3,2), (1,3), (2,4), (1,4)(2,3), (1,2)(3,4) \}</math>

This can be related to the geometric definition by thinking of <math>1,2,3,4</math> as the vertices of the square and considering an element of <math>D_8</math> in terms of its induced action on the vertices. It relates to the presentation via setting <math>a = (1,2,3,4)</math> and <math>x = (1,3)</math>.

===Multiplication table===

Here, <math>e</math> denotes the identity element, <math>a</math> is an element of order 4, and <math>x</math> is an element of order two that isn't equal to <math>a^2</math>, as in the above presentation. 

{{#lst:element structure of dihedral group:D8|multiplication table}}

===Other definitions===

The dihedral group can be described in the following ways:

# The [[dihedral group]] of order eight.
# The [[generalized dihedral group]] corresponding to [[cyclic group:Z4|the cyclic group of order four]].
# The [[holomorph of a group|holomorph]] of the [[cyclic group:Z4|cyclic group of order four]].
# The [[external wreath product]] of the cyclic group of order two with the cyclic group of order two, acting via the regular action.
# The <math>2</math>-[[Sylow subgroup]] of the [[symmetric group:S4|symmetric group on four letters]].
# The <math>2</math>-[[Sylow subgroup]] of the [[symmetric group:S5|symmetric group on five letters]].
# The <math>2</math>-[[Sylow subgroup]] of the [[alternating group:A6|alternating group on six letters]].
# The [[member of family::unitriangular matrix group of degree three]] <math>UT(3,2)</math> over [[field:F2]], <math>2</math>-[[Sylow subgroup]] of [[projective special linear group:PSL(3,2)|PSL(3,2)]].
# The [[member of family::extraspecial group]] of order <math>2^3</math> and type '+'.

==Position in classifications==

{| class="sortable" border="1"
! Type of classification !! Name in that classification
|-
| GAP ID || (8,3), i.e., the third among the groups of order 8
|-
| Hall-Senior number || (8,4), i.e., 4 among groups of order 8
|-
| Hall-Senior symbol || <math>8\Gamma_2a_1</math> 
|}

==Elements==

{{further|[[element structure of dihedral group:D8]]}}
{{#lst:element structure of dihedral group:D8|elements}}

Below is the conjugacy and automorphism class structure:

{{#lst:element structure of dihedral group:D8|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 for function value
|-
| [[underlying prime of p-group]] || [[arithmetic function value::underlying prime of p-group;2|2]] || || 
|-
| {{arithmetic function value order|8}} || As a semidirect product of <math>\Z_4</math> and <math>\Z_2</math>: the order is the product of the orders of <math>\Z_4</math> and <math>\Z_2</math>, which is <math>4 \times 2 = 8</math> <br><br> As a wreath product of <math>\Z_2</math> and <math>\Z_2</math>: the order is <math>2^2 \cdot 2 = 8</math>
|-
| {{arithmetic function value order p-log etc|3}}
|-
| {{arithmetic function value exponent given order|4|8}} || As a dihedral group: the dihedral group of order <math>2n</math> has exponent equal to <math>\operatorname{lcm} \{ n,2 \}</math>.
|-
| [[prime-base logarithm of exponent]] || [[arithmetic function value::prime-base logarithm of exponent;2|2]] || ||
|-
| {{arithmetic function value given order and p-log|nilpotency class|2|8|3}} || The [[derived subgroup]] is <math>\{ e, a^2 \}</math> -- and it is the same as the [[center]]. See [[center of dihedral group:D8]]. Also see [[element structure of dihedral group:D8#Commutator map]]
|-
| {{arithmetic function value given order and p-log|derived length|2|8|3}} || The [[derived subgroup]] is <math>\{ e, a^2 \}</math>, which is abelian. See [[center of dihedral group:D8]].
|-
| {{arithmetic function value given order and p-log|Frattini length|2|8|3}} ||  The [[Frattini subgroup]] is <math>\{ e, a^2 \}</math>, which is of prime order, hence its Frattini subgroup is trivial.
|-
| [[Fitting length]] || [[arithmetic function value::Fitting length;1|1]] || || All groups of prime power order are nilpotent, hence have Fitting length 1.
|-
| {{arithmetic function value given order and p-log|minimum size of generating set|2|8|3}} || Generator of cyclic subgroup of order four and element of order two outside.
|-
| {{arithmetic function value given order and p-log|subgroup rank of a group|2|8|3}} || All proper subgroups are cyclic or [[Klein four-group]]s.
|-
| {{arithmetic function value given order and p-log|rank of a p-group|2|8|3}} || There exist Klein four-subgroups.
|-
| {{arithmetic function value given order and p-log|normal rank of a p-group|2|8|3}} || There exist normal Klein four-subgroups.
|-
| {{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 dihedral group:D8]]}}

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation for function value !! GAP verification (set <tt>G := DihedralGroup(8);</tt>) -- See more at [[#GAP implementation]]
|-
| {{arithmetic function value given order|number of conjugacy classes|5|8|3}} || As dihedral group <math>D_{2n}</math>, <math>n</math> even: <br><math>\! (n + 6)/2 = (4 + 6)/2 = 5</math>. See [[element structure of dihedral groups]] and [[element structure of dihedral group:D8]]<br>As unitriangular matrix group <math>UT(3,q), q = 2</math>:<br><math>q^2 + q - 1 = 2^2 + 2 - 1 = 5</math><br>See [[element structure of unitriangular matrix group of degree three over a finite field]] || {{GAP verify list length|ConjugacyClasses}}
|-
| {{arithmetic function value given order|number of equivalence classes under real conjugacy|5|8|3}} || Same as number of conjugacy classes, because the group is an [[ambivalent group]]. See [[dihedral groups are ambivalent]] || 
|-
| {{arithmetic function value given order|number of conjugacy classes of real elements|5|8|3}} || Same as number of conjugacy classes, because the group is an [[ambivalent group]]. See [[dihedral groups are ambivalent]] ||
|-
| {{arithmetic function value given order|number of equivalence classes under rational conjugacy|5|8|3}} || Same as number of conjugacy classes, because the group is a [[rational group]]. || {{GAP verify list length|RationalClasses}}
|-
| {{arithmetic function value given order|number of conjugacy classes of rational elements|5|8|3}} || Same as number of conjugacy classes, because the group is a [[rational group]].  || 
|}

===Arithmetic functions of a subgroup-counting nature===

{{further|[[subgroup structure of dihedral group:D8]]}}

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation !! GAP verification (set <tt>G := DihedralGroup(8);</tt>) -- See more at [[#GAP verification]]
|-
| {{arithmetic function value|number of subgroups|10}} ||  || As a dihedral group <math>\! D_{2n}, n = 4</math> number of subgroups is <math>\! d(n) + \sigma(n) = d(4) + \sigma(4) = 3 + 7 = 10</math>, where <math>d</math> is the divisor count function and <math>\sigma</math> is the divisor sum function. See [[subgroup structure of dihedral group:D8]], [[subgroup structure of dihedral groups]] || {{GAP verify list length|Subgroups}}
|-
| {{arithmetic function value|number of conjugacy classes of subgroups|8}} ||  || See [[subgroup structure of dihedral groups]], [[subgroup structure of dihedral group:D8]] || {{GAP verify list length|ConjugacyClassesSubgroups}}
|-
| {{arithmetic function value given order|number of normal subgroups|6|8}} || See [[subgroup structure of dihedral groups]], [[subgroup structure of dihedral group:D8#Lattice of normal subgroups]] || {{GAP verify list length|NormalSubgroups}}
|-
| {{arithmetic function value|number of automorphism classes of subgroups|6}} || || ||
|-
| {{arithmetic function value|number of characteristic subgroups|4}} || || || {{GAP verify list length|CharacteristicSubgroups}}
|}

===Lists of numerical invariants===

{| class="sortable" border="1"
! List !! Value !! Explanation/comment
|-
| [[conjugacy class size set|conjugacy class sizes]] || 1,1,2,2,2 || Two central elements, all others in conjugacy classes of size two. See [[element structure of dihedral group:D8]] and [[element structure of dihedral groups]].
|-
| sizes of orbits under automorphism group || 1,1,2,4 || Two central elements, one conjugacy class of elements of order four, one orbit of size four, comprising two conjugacy classes of size, with all elements non-central of order two.
|-
| [[order statistics]] || <math>1 \mapsto 1, 2 \mapsto 5, 4 \mapsto 2</math> || Of the five elements of order two, one is central. The other four are automorphic to each other. See [[element structure of dihedral group:D8]] and [[element structure of dihedral groups]]
|-
| [[degrees of irreducible representations]] || <math>1,1,1,1,2</math> || See [[linear representation theory of dihedral group:D8]]
|-
| orders of subgroups || <math>1,2,2,2,2,2,4,4,4,8</math> || See [[subgroup structure of dihedral group:D8]]
|}

===Action-based/automorphism group realization invariants===

{| class="sortable" border="1"
! Function !! Value !! Explanation
|-
| minimum degree of faithful representation || 2 || 
|-
| minimum degree of nontrivial irreducible representation || 2 ||
|-
| smallest size of set with faithful action || 4 ||
|-
| smallest size of set with faithful transitive action || 4 ||
|-
| [[symmetric genus]] || ? ||
|}

==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>a</math> and <math>x</math> don't commute || Smallest non-abelian [[satisfies property::group of prime power order]]
|-
|[[Dissatisfies property::T-group]] || No || <math>\langle x \rangle \triangleleft \langle a^2,x \rangle</math>, which is normal, but <math>\langle x \rangle</math> is not normal || Smallest example for [[normality is not transitive]].
|-
|[[Satisfies property::monolithic group]] || Yes|| Unique minimal normal subgroup of order two || 
|}

===Other properties===

{| class="sortable" border="1"
!Property !! Satisfied? !! Explanation !! Comment
|-
|[[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::algebra group]] || Yes || It is isomorphic to the [[unitriangular matrix group of degree three]] over [[field:F2]], which is clearly an algebraic group. ||
|-
| [[Satisfies property::ambivalent group]] || Yes || [[dihedral groups are ambivalent]] || Also see [[generalized dihedral groups are ambivalent]]
|-
|[[Satisfies property::rational group]] || Yes || Any two elements that generate the same cyclic group are conjugate || Thus, all characters are integer-valued.
|-
|[[Satisfies property::rational-representation group]] || Yes || All representations over characteristic zero are realized over the rationals. || Contrast with [[quaternion group]], that is rational but not rational-representation.
|-
| {{group properties because extraspecial}}
|-
| [[Satisfies property::maximal class 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|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|any two elements generating the same cyclic subgroup are automorphic]] || Yes || ||
|-
|[[Dissatisfies property::group in which every element is order-automorphic|every element is order-automorphic]] || No || ||
|-
|[[Satisfies property::directly indecomposable group]] || Yes || ||
|-
|[[Satisfies property::centrally indecomposable group]] || Yes || ||
|-
|[[Dissatisfies property::splitting-simple group]] || No || ||
|-
|[[Satisfies property::stem group]] || Yes || the center equals the derived subgroup, and hence, in particular, is contained in the derived subgroup. ||
|-
|[[Dissatisfies property::Schur-trivial group]] || No || the [[Schur multiplier]] is [[cyclic group:Z2]]; see [[group cohomology of dihedral group:D8]]. ||
|}

==Subgroups==
{{further|[[subgroup structure of dihedral group:D8]]}}

[[File:D8latticeofsubgroups.png|800px|Lattice of subgroups of the dihedral group]]

{{#lst:subgroup structure of dihedral group:D8|summary}}

===Subgroup-defining functions and associated quotient-defining functions===

{{further|[[subgroup structure of dihedral group:D8#Defining functions]]}}
{{#lst:subgroup structure of dihedral group:D8|sdf summary}}

==Automorphisms and endomorphisms==

{{further|[[Endomorphism structure of dihedral group:D8]]}}
{{#lst:endomorphism structure of dihedral group:D8|summary}}

==Linear representation theory==

{{further|[[Linear representation theory of dihedral group:D8]], [[linear representation theory of dihedral groups]]}}

===Summary===

{{#lst:linear representation theory of dihedral group:D8|summary}}

===Character table===

{{#lst:linear representation theory of dihedral group:D8|character table}}
==Fusion systems==

{{further|[[fusion systems for dihedral group:D8]]}}
===Summary===
{{#lst:fusion systems for dihedral group:D8|summary}}

===Description of fusion systems===
{{#lst:fusion systems for dihedral group:D8|description}}

==Distinguishing features==

===Smallest of its kind===

* This is the unique non-[[T-group]] of smallest order, i.e., the unique smallest example of a group in which [[normality is not transitive]].
* This is a non-abelian nilpotent group of smallest order, though not the only one. The other such group is the [[quaternion group]].

===Different from others of the same order===

* It is the only group of its order that is isomorphic to its [[automorphism group]].
* It is the only group of its order that is not a [[T-group]].
* It is the only group of its order having two Klein four-subgroups. In particular, it gives an example of a situation where the number of elementary abelian subgroups of order <math>p^2</math> is neither zero nor <math>1</math> modulo <math>p</math>. Contrast this with the case of odd <math>p</math>, where we have the [[congruence condition on number of elementary abelian subgroups of prime-square order for odd prime]].

==GAP implementation==
{{access GAP implementation online using SAGE|id=5013}}
{{GAP ID|8|3}}
{{HallSenior|8|4}}
===Short descriptions===

{| class="sortable" border="1"
! Description !! GAP functions used !! Mathematical translation of description
|-
| <tt>DihedralGroup(8)</tt> || [[GAP:DihedralGroup|DihedralGroup]] || dihedral group of order <math>8</math>, degree <math>4</math>
|-
| <tt>WreathProduct(CyclicGroup(2),CyclicGroup(2))</tt> || [[GAP:WreathProduct|WreathProduct]], [[GAP:CyclicGroup|CyclicGroup]] || [[external wreath product]] of two copies of cyclic group of order two
|-
| <tt>ExtraspecialGroup(2^3,'+')</tt> || [[GAP:ExtraspecialGroup|ExtraspecialGroup]] || [[extraspecial group]] of '+' type for the prime <math>2</math> and order <math>2^3</math>
|-
| <tt>SylowSubgroup(SymmetricGroup(4),2)</tt> || [[GAP:SylowSubgroup|SylowSubgroup]] and [[GAP:SymmetricGroup|SymmetricGroup]] || The <math>2</math>-[[Sylow subgroup]] of the [[symmetric group:S4|symmetric group of degree four]]
|-
| <tt>SylowSubgroup(GL(3,2),2)</tt> || [[GAP:SylowSubgroup|SylowSubgroup]], [[GAP:GL|GL]] || The <math>2</math>-Sylow subgroup of [[GL(3,2)]]
|}

===Description by presentation===

Here is the code:

<pre>gap> F := FreeGroup(2);;
gap> G := F/[F.1^4, F.2^2, F.2 * F.1 * F.2 * F.1];
<fp group on the generators [ f1, f2 ]>
gap> IdGroup(G);
[ 8, 3 ]</pre>

The group <math>G</math> constructed here is the dihedral group of order <math>8</math>. The first generator <math>F.1</math> maps to the ''rotation'' element of order four and the second generator <math>F.2</math> maps to a ''reflection'' element of order two.

===Long descriptions===

It can be described as the [[holomorph of a group|holomorph]] of the cyclic group of order four. For this, first define <math>C</math> to be the cyclic group of order four (using [[GAP:CyclicGroup|CyclicGroup]]), and then use [[GAP:SemidirectProduct|SemidirectProduct]] and [[GAP:AutomorphismGroup|AutomorphismGroup]]:

<pre>C := CyclicGroup(4);
G := SemidirectProduct(AutomorphismGroup(C),C);</pre>

Here, <math>G</math> is the dihedral group of order eight. We can also construct it as a semidirect product of the Klein four-group and an automorphism of order two.

<pre>K := DirectProduct(CyclicGroup(2),CyclicGroup(2));
A := AutomorphismGroup(K);
S := SylowSubgroup(A,2);
G := SemidirectProduct(S,K);</pre>

Then, <math>G</math> is isomorphic to the dihedral group of order eight.

===GAP verification===

Below is a GAP implementation verifying the various function values and group properties as stated in this page. Before beginning, set <tt>G := DihedralGroup(8);</tt> or any equivalent way of setting <math>G</math> to be dihedral of order eight.

<pre>gap> IdGroup(G);
[ 8, 3 ]
gap> Order(G);
8
gap> Exponent(G);
4
gap> NilpotencyClassOfGroup(G);
2</pre>

More: <toggledisplay>

<pre>gap> DerivedLength(G);
2
gap> FrattiniLength(G);
2
gap> Rank(G);
2
gap> SubgroupRank(G);
2
gap> RankAsPGroup(G);
2
gap> NormalRank(G);
2
gap> CharacteristicRank(G);
1
gap> Length(ConjugacyClasses(G));
5
gap> Length(RationalClasses(G));
5
gap> Length(Subgroups(G));
10
gap> Length(ConjugacyClassesSubgroups(G));
8
gap> Length(NormalSubgroups(G));
6
gap> Length(CharacteristicSubgroups(G));
4</pre></toggledisplay>