Difference between revisions of "Dihedral group"

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==Definition==
 
==Definition==
  
The '''dihedral group''' with parameter <math>n</math>, denoted sometimes as <math>D_n</math> and sometimes as <math>D_{2n}</math> (this wiki uses <math>D_{2n}</math>) is defined in the following equivalent ways:
+
The '''dihedral group''' of degree <math>n</math> and order <math>2n</math>, denoted sometimes as sometimes as <math>D_{2n}</math> ('''this wiki uses <math>D_{2n}</math>'''), sometimes as <math>D_n</math>, and sometimes as <math>\operatorname{Dih}_n</math>, is defined in the following equivalent ways:
  
* It has the [[presentation]]:
+
* It has the [[presentation]] (here, <math>e</math> denotes the identity element):
  
 
<math>\langle x,a \mid a^n = x^2 = e, xax^{-1} = a^{-1} \rangle</math>
 
<math>\langle x,a \mid a^n = x^2 = e, xax^{-1} = a^{-1} \rangle</math>
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<math>\langle x,a \mid x^2 = e, xax^{-1} = a^{-1} \rangle</math>.
 
<math>\langle x,a \mid x^2 = e, xax^{-1} = a^{-1} \rangle</math>.
  
==Elements==
+
==Related notions==
  
===In the case of twice an odd number===
+
===Generalizations===
  
Suppose <math>n</math> is odd and greater than <math>1</math>. Let <math>D_{2n}</math> be the dihedral group of order <math>2n</math>. It has the following elements:
+
* [[Generalized dihedral group]]: This is a semidirect product of an abelian group by a cyclic group of order two acting via the inverse map.
 +
* [[q-hedral group]]
 +
* [[Coxeter group]]: Dihedral groups are Coxeter groups with two generators.
  
* <math>n</math> elements of odd order, all of them in the cyclic subgroup <math>\langle a \rangle</math> of order <math>n</math>. Of these, there are precisely <math>\varphi(d)</math> elements of order <math>d</math> for any divisor <math>d</math> of <math>n</math>.
+
==Particular cases==
* <math>n</math> elements of order <math>2</math>. These are the elements outside the cyclic subgroup <math>\langle a \rangle</math>.
 
  
The conjugacy classes of elements are as follows (a total of <math>(n+3)/2</math> conjugacy classes):
+
===For small values===
  
* The identity element is its own conjugacy class.
+
Note that all dihedral groups are [[metacyclic group|metacyclic]] and hence supersolvable. A dihedral group is nilpotent if and only if it is of order <math>2^k</math> for some <math>k</math>. It is abelian only if it has order <math>2</math> or <math>4</math>.
* The non-identity elements in <math>\langle a \rangle</math> occur in conjugacy classes of size two. Each element is conjugate in <math>D_{2n}</math> to its inverse. Thus, these form <math>(n-1)/2</math> conjugacy classes.
 
* The elements outside <math>\langle a \rangle</math> all form a single conjugacy class of size <math>n</math>.
 
  
The equivalence classes of elements upto automorphism are as follows:
+
{| class="sortable" border="1"
 +
! Order of group !! Degree (size of regular polygon it acts on) !! Common name for the group !! Comment
 +
|-
 +
| 2 || 1 || [[Cyclic group:Z2]] || Not usually considered a dihedral group.
 +
|-
 +
| 4 || 2 || [[Klein four-group]] || [[elementary abelian group]] that is not [[cyclic group|cyclic]]
 +
|-
 +
| 6 || 3 || [[symmetric group:S3]] || [[metacyclic group|metacyclic]], hence [[supersolvable group|supersolvable]] but not [[nilpotent group|nilpotent]]
 +
|-
 +
| 8 || 4 || [[dihedral group:D8]] || [[nilpotent group|nilpotent]] but not [[abelian group|abelian]]
 +
|-
 +
| 10 || 5 || [[dihedral group:D10]] || [[metacyclic group|metacyclic]], hence [[supersolvable group|supersolvable]] but not [[nilpotent group|nilpotent]]
 +
|-
 +
|12 || 6 || [[dihedral group:D12]] || direct product of the dihedral group of order six and the cyclic group of order two.
 +
|-
 +
| 16 || 8 || [[dihedral group:D16]] || [[nilpotent group|nilpotent]] but not [[abelian group|abelian]]
 +
|}
  
* Two elements inside <math>\langle a \rangle</math> are related via an automorphism if and only if they generate the same cyclic subgroup.
+
==Arithmetic functions==
* Any two elements outside <math>\langle a \rangle</math> are related via an automorphism (in fact, they are in the same [[conjugacy class]]).
 
  
===In the case of twice an even number===
+
<math>n</math> here denotes the ''degree'', or half the order, of the dihedral group, which we denote as <math>D_{2n}</math>.
  
Suppose <math>n = 2m</math>, and <math>D_{2n}</math> is the dihedral group of order <math>2n</math>. Then, <math>D_{2n}</math> has the following conjugacy classes (a total of <math>(n + 6)/2</math> conjugacy classes):
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{| class="sortable" border="1"
 +
! Function !! Value !! Explanation
 +
|-
 +
| [[Order of a group|order]] || <math>2n</math> || Cyclic subgroup of order <math>n</math>, quotient of order <math>2</math>.
 +
|-
 +
| [[Exponent of a group|exponent]] || <math>\operatorname{lcm}(n,2)</math> || Exponent of cyclic subgroup is <math>n</math>, elements outside have order <math>2</math>.
 +
|-
 +
| [[derived length]] || <math>2</math> for <math>n \ge 3</math>, <math>1</math> for <math>n = 1,2</math> ||
 +
|-
 +
| [[nilpotency class]] || <math>k</math> when <math>n = 2^k</math>, none otherwise || Nilpotent only if <math>n</math> is a power of <math>2</math>.
 +
|-
 +
| [[max-length of a group|max-length]] || 1 + sum of exponents of prime divisors of <math>n</math> || The dihedral groups are solvable.
 +
|-
 +
| [[composition length]] || 1 + sum of exponents of prime divisors of <math>n</math> || The dihedral groups are solvable.
 +
|-
 +
| [[number of subgroups]] || <math>\sigma(n) + d(n)</math> || <math>\sigma(n)</math> is the [[number:divisor sum function|divisor sum function]] and <math>d(n)</math> is the [[number:divisor count function|divisor count function]].
 +
|-
 +
| [[number of conjugacy classes]] || <math>(n+3)/2</math> if <math>n</math> is odd, <math>(n+6)/2</math> if <math>n</math> is even. ||
 +
|-
 +
| [[number of conjugacy classes of subgroups]] || <math>3d(n) - d(m)</math> where <math>m</math> is the largest odd divisor of <math>n</math> || <math>d(n)</math> subgroups inside the cyclic part. For odd divisors, one external conjugacy class of subgroups per divisor; for even divisors, one external conjugacy class per divisor.
 +
|}
  
* There are <math>(n+2)/2</math> conjugacy classes inside <math>\langle a \rangle</math>: The identity element and <math>a^m</math> are both central elements. All other elements have a conjugacy class of size two: the element and its inverse.
+
==Group properties==
* There are two conjugacy classes outside <math>a</math>, of size <math>n/2</math> each.
+
 
 +
{| class="sortable" border="1"
 +
!Property !! Satisfied? !! Explanation
 +
|-
 +
|[[Abelian group]] || no for <math>n \ge 3</math> || For <math>n \ge 3</math>, the elements <math>a,x</math> do not commute.
 +
|-
 +
|[[Nilpotent group]] || yes for <math>n</math> a power of <math>2</math>, no otherwise ||
 +
|-
 +
|[[Metacyclic group]] || yes ||
 +
|-
 +
|[[Supersolvable group]] || yes ||
 +
|-
 +
|[[Solvable group]] || yes ||
 +
|-
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|[[T-group]] || yes for <math>n</math> odd or twice an odd number, no for <math>n</math> a multiple of <math>4</math> ||
 +
|-
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|[[Rational group]] || yes for <math>n \le 4</math> and <math>n = 6</math>, no for <math>n = 5</math>, <math>n \ge 7</math> || The field generated by character values is <math>\mathbb{Q}(\cos(2\pi / n))</math>. The number <math>\cos(2\pi/n)</math> is rational if and only if <math>n = 1, 2, 3, 4, 6</math>. This can be proved by showing that <math>2\cos(2\pi/n)</math> is an algebraic integer (it is the root of a monic polynomial with integer coefficients) and therefore must be in <math>\{ -2, -1, 0, 1, 2 \}</math>, which translates to these five values of <math>n</math>. For more information, see [[linear representation theory of dihedral groups]].
 +
|-
 +
|[[Rational-representation group]] || yes for <math>n \le 4</math> and <math>n = 6</math>, no for <math>n = 5</math>, <math>n \ge 7</math> || The unique minimal splitting field is <math>\mathbb{Q}(\cos(2\pi / n))</math>. The number <math>\cos(2\pi/n)</math> is rational if and only if <math>n = 1, 2, 3, 4, 6</math>. This can be proved by showing that <math>2\cos(2\pi/n)</math> is an algebraic integer (it is the root of a monic polynomial with integer coefficients) and therefore must be in <math>\{ -2, -1, 0, 1, 2 \}</math>, which translates to these five values of <math>n</math>. For more information, see [[linear representation theory of dihedral groups]].
 +
|-
 +
|[[Ambivalent group]] || yes || Elements in the cyclic subgroup are conjugate via <math>x</math>, elements outside have order two.
 +
|}
 +
 
 +
==Elements==
 +
 
 +
{{further|[[element structure of dihedral groups]]}}
 +
===Summary===
 +
{{#lst:element structure of dihedral groups|summary}}
 +
===Conjugacy class structure===
 +
{{#lst:element structure of dihedral groups|conjugacy class structure}}
  
 
==Subgroups==
 
==Subgroups==
 +
 +
{{further|[[Subgroup structure of dihedral groups]]}}
  
 
There are two kinds of subgroups:
 
There are two kinds of subgroups:
  
* Subgroups of the form <math>\langle a^d \rangle</math>, where <math>d|n</math>. The number of such subgroups equals the number of positive divisors of <math>n</math>, sometimes denoted <math>\tau(n)</math>. The subgroup generated by <math>a^d</math> is a [[cyclic group]] of order <math>n/d</math>.
+
* Subgroups of the form <math>\langle a^d \rangle</math>, where <math>d|n</math>. The number of such subgroups equals the [[number:divisor count function|number of positive divisors]] of <math>n</math>, sometimes denoted <math>\tau(n)</math>. The subgroup generated by <math>a^d</math> is a [[cyclic group]] of order <math>n/d</math>.
* Subgroups of the form <math>\langle a^d, a^r x \rangle</math>, where <math>d|n</math> and <math>0 \le r < d</math>. The number of such subgroups equals the sum of all positive divisors of <math>n</math>, sometimes denoted <math>\sigma(n)</math>. The subgroup of the above form is a [[dihedral group]] of order <math>2n/d</math>.
+
* Subgroups of the form <math>\langle a^d, a^r x \rangle</math>, where <math>d|n</math> and <math>0 \le r < d</math>. The number of such subgroups equals the [[number:divisor sum function|sum of all positive divisors]] of <math>n</math>, sometimes denoted <math>\sigma(n)</math>. The subgroup of the above form is a [[dihedral group]] of order <math>2n/d</math>.
  
 
In particular, all subgroups of the dihedral group are either cyclic or dihedral.
 
In particular, all subgroups of the dihedral group are either cyclic or dihedral.
  
 
Also note that the dihedral group has subgroups of all orders dividing its order. This is true more generally for all [[finite supersolvable group]]s. {{further|[[Finite supersolvable implies subgroups of all orders dividing the group order]]}}
 
Also note that the dihedral group has subgroups of all orders dividing its order. This is true more generally for all [[finite supersolvable group]]s. {{further|[[Finite supersolvable implies subgroups of all orders dividing the group order]]}}
 +
 +
==Linear representation theory==
 +
 +
{{further|[[linear representation theory of dihedral groups]]}}
 +
 +
{{#lst:linear representation theory of dihedral groups|summary}}
  
 
==Supergroups==
 
==Supergroups==
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Dicyclic groups whose order is a power of <math>2</math> are termed [[generalized quaternion group]]s.
 
Dicyclic groups whose order is a power of <math>2</math> are termed [[generalized quaternion group]]s.
  
==Particular cases==
+
==Internal links==
  
===For small values===
+
* [[Element structure of dihedral groups]]
 +
* [[Subgroup structure of dihedral groups]]
 +
* [[Linear representation theory of dihedral groups]]
 +
* [[Group cohomology of dihedral groups]]
  
Note that all dihedral groups are [[metacyclic group|metacyclic]] and hence supersolvable. A dihedral group is nilpotent if and only if it is of order <math>2^k</math> for some <math>k</math>. It is abelian only if it has order <math>2</math> or <math>4</math>.
 
 
{| class="wikitable" border="1"
 
! Order of group !! Size of regular polygon it acts on !! Common name for the group !! Comment
 
|-
 
| 4 || 2 || [[Klein four-group]] || [[elementary abelian group]] that is not [[cyclic group|cyclic]]
 
|-
 
| 6 || 3 || [[symmetric group:S3]] || [[metacyclic group|metacyclic]], hence [[supersolvable group|supersolvable]] but not [[nilpotent group|nilpotent]]
 
|-
 
| 8 || 4 || [[dihedral group:D8]] || [[nilpotent group|nilpotent]] but not [[abelian group|abelian]]
 
|-
 
| 10 || 5 || [[dihedral group:D10]] || [[metacyclic group|metacyclic]], hence [[supersolvable group|supersolvable]] but not [[nilpotent group|nilpotent]]
 
|}
 
 
==References==
 
==References==
 
===Textbook references===
 
===Textbook references===

Latest revision as of 02:22, 6 July 2019

WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with dicyclic group (also called binary dihedral group)
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This is a family of groups parametrized by the natural numbers, viz, for each natural number, there is a unique group (upto isomorphism) in the family corresponding to the natural number. The natural number is termed the parameter for the group family

Definition

The dihedral group of degree n and order 2n, denoted sometimes as sometimes as D_{2n} (this wiki uses D_{2n}), sometimes as D_n, and sometimes as \operatorname{Dih}_n, is defined in the following equivalent ways:

  • It has the presentation (here, e denotes the identity element):

\langle x,a \mid a^n = x^2 = e, xax^{-1} = a^{-1} \rangle

  • (For n \ge 3): It is the group of symmetries of a regular n-gon in the plane, viz., the plane isometries that preserves the set of points of the regular n-gon.

The dihedral groups arise as a special case of a family of groups called von Dyck groups. They also arise as a special case of a family of groups called Coxeter groups.

Note that for n = 1 and n = 2, the geometric description of the dihedral group does not make sense. In these cases, we use the algebraic description.

The infinite dihedral group, which is the n = \infty case of the dihedral group and is denoted D_\infty and is defined as:

\langle x,a \mid x^2 = e, xax^{-1} = a^{-1} \rangle.

Related notions

Generalizations

Particular cases

For small values

Note that all dihedral groups are metacyclic and hence supersolvable. A dihedral group is nilpotent if and only if it is of order 2^k for some k. It is abelian only if it has order 2 or 4.

Order of group Degree (size of regular polygon it acts on) Common name for the group Comment
2 1 Cyclic group:Z2 Not usually considered a dihedral group.
4 2 Klein four-group elementary abelian group that is not cyclic
6 3 symmetric group:S3 metacyclic, hence supersolvable but not nilpotent
8 4 dihedral group:D8 nilpotent but not abelian
10 5 dihedral group:D10 metacyclic, hence supersolvable but not nilpotent
12 6 dihedral group:D12 direct product of the dihedral group of order six and the cyclic group of order two.
16 8 dihedral group:D16 nilpotent but not abelian

Arithmetic functions

n here denotes the degree, or half the order, of the dihedral group, which we denote as D_{2n}.

Function Value Explanation
order 2n Cyclic subgroup of order n, quotient of order 2.
exponent \operatorname{lcm}(n,2) Exponent of cyclic subgroup is n, elements outside have order 2.
derived length 2 for n \ge 3, 1 for n = 1,2
nilpotency class k when n = 2^k, none otherwise Nilpotent only if n is a power of 2.
max-length 1 + sum of exponents of prime divisors of n The dihedral groups are solvable.
composition length 1 + sum of exponents of prime divisors of n The dihedral groups are solvable.
number of subgroups \sigma(n) + d(n) \sigma(n) is the divisor sum function and d(n) is the divisor count function.
number of conjugacy classes (n+3)/2 if n is odd, (n+6)/2 if n is even.
number of conjugacy classes of subgroups 3d(n) - d(m) where m is the largest odd divisor of n d(n) subgroups inside the cyclic part. For odd divisors, one external conjugacy class of subgroups per divisor; for even divisors, one external conjugacy class per divisor.

Group properties

Property Satisfied? Explanation
Abelian group no for n \ge 3 For n \ge 3, the elements a,x do not commute.
Nilpotent group yes for n a power of 2, no otherwise
Metacyclic group yes
Supersolvable group yes
Solvable group yes
T-group yes for n odd or twice an odd number, no for n a multiple of 4
Rational group yes for n \le 4 and n = 6, no for n = 5, n \ge 7 The field generated by character values is \mathbb{Q}(\cos(2\pi / n)). The number \cos(2\pi/n) is rational if and only if n = 1, 2, 3, 4, 6. This can be proved by showing that 2\cos(2\pi/n) is an algebraic integer (it is the root of a monic polynomial with integer coefficients) and therefore must be in \{ -2, -1, 0, 1, 2 \}, which translates to these five values of n. For more information, see linear representation theory of dihedral groups.
Rational-representation group yes for n \le 4 and n = 6, no for n = 5, n \ge 7 The unique minimal splitting field is \mathbb{Q}(\cos(2\pi / n)). The number \cos(2\pi/n) is rational if and only if n = 1, 2, 3, 4, 6. This can be proved by showing that 2\cos(2\pi/n) is an algebraic integer (it is the root of a monic polynomial with integer coefficients) and therefore must be in \{ -2, -1, 0, 1, 2 \}, which translates to these five values of n. For more information, see linear representation theory of dihedral groups.
Ambivalent group yes Elements in the cyclic subgroup are conjugate via x, elements outside have order two.

Elements

Further information: element structure of dihedral groups

Summary

Item Value
order 2n
conjugacy class sizes Case n odd: 1 (1 time), 2 ((n-1)/2 times), n (1 time)
Case n even: 1 (2 times), 2 ((n-2)/2 times), n/2 (2 times)
number of conjugacy classes (n + 3)/2 if n odd, (n + 6)/2 if n even
number of orbits under automorphism group d(n) + 1 where d is the divisor count function if n \ge 3, 2 if n = 2
order statistics Case n odd: \varphi(d) of order d for d | n, n of order 2

Conjugacy class structure

Nature of conjugacy class Size of each conjugacy class Number of such conjugacy classes Total number of elements
Identity element 1 1 1
Non-identity elements in cyclic subgroup \langle a \rangle, where each element and its inverse form a conjugacy class 2 (n - 1)/2 n - 1
Elements outside the cyclic subgroup \langle a \rangle, all form a single conjugacy class n 1 n
Total -- (n + 3)/2 (number of conjugacy classes) 2n (order of group)


Subgroups

Further information: Subgroup structure of dihedral groups

There are two kinds of subgroups:

In particular, all subgroups of the dihedral group are either cyclic or dihedral.

Also note that the dihedral group has subgroups of all orders dividing its order. This is true more generally for all finite supersolvable groups. Further information: Finite supersolvable implies subgroups of all orders dividing the group order

Linear representation theory

Further information: linear representation theory of dihedral groups


Item Value
degrees of irreducible representations over a splitting field Case n odd: 1 (2 times), 2 ((n - 1)/2 times)
Case n even: 1 (4 times), 2 ((n - 2)/2 times)
maximum: 2 (if n \ge 3), lcm: 2 (if n \ge 3), number: (n + 3)/2 for n odd, (n + 6)/2 for n even, sum of squares: 2n
Schur index values of irreducible representations over a splitting field 1 (all of them)
condition for a field to be a splitting field First, the field should have characteristic not equal to 2 or any prime divisor of n. Also, take the cyclotomic polynomial \Phi_n(x). Let \zeta be a root of the polynomial. Then, the field should contain the element \zeta + \zeta^{-1}, i.e., the minimal polynomial for \zeta + \zeta^{-1} should split completely.
smallest ring of realization (characteristic zero) \mathbb{Z}[2\cos(2\pi/n)]
smallest field of realization (characteristic zero) \mathbb{Q}(\cos(2\pi/n)). Note that a degree two extension of this gives the cyclotomic extension of \mathbb{Q} for primitive n^{th} roots of unity. The given field can be thought of as the intersection of the cyclotomic extension and the real numbers.
smallest size splitting field unclear. Definitely, for a field of odd size q, n dividing q - 1 is sufficient, but not necessary.
degrees of irreducible representations over rational numbers PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]


Supergroups

Groups having the dihedral group as quotient

The dicyclic group, also called the binary dihedral group, of order 4n, has the dihedral group of order 2n as a quotient -- in fact the quotient by its center, which is a cyclic subgroup of order two. the presentation for the dicyclic group is given by:

\langle a,x \mid a^n = x^2 = (ax)^2 \rangle.

Dicyclic groups whose order is a power of 2 are termed generalized quaternion groups.

Internal links

References

Textbook references

  • Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347More info, Page 23-27, Section 1.2 Dihedral Groups (the entire section discusses dihedral groups from a number of perspectives)
  • Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, More info, Page 24 (definition introduced in paragraph)
  • Algebra by Serge Lang, ISBN 038795385X, More info, Page 78, Exercise 34 (a) (definition introduced in exercise)
  • Topics in Algebra by I. N. Herstein, More info, Page 54, Problem 17
  • A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, More info, Page 6 (definition introduced informally, in paragraph, using the geometric perspective)
  • An Introduction to Abstract Algebra by Derek J. S. Robinson, ISBN 3110175444, More info, Page 42, under The symmetry group of the regular n-gon
  • Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189, More info, Page 50 (definition introduced as a subgroup of the symmetric group)