Difference between revisions of "Dihedral group"
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* [[Coxeter group]]: Dihedral groups are Coxeter groups with two generators. | * [[Coxeter group]]: Dihedral groups are Coxeter groups with two generators. | ||
− | == | + | ==Particular cases== |
+ | |||
+ | ===For small values=== | ||
+ | |||
+ | 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 !! 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]] | ||
+ | |} | ||
+ | |||
+ | ==Arithmetic functions== | ||
+ | |||
+ | <math>n</math> here denotes the ''degree'', or half the order, of the dihedral group, which we denote as <math>D_{2n}</math>. | ||
+ | |||
+ | {| class="wikitable" 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. || | ||
+ | |} | ||
+ | |||
+ | ==Group properties== | ||
+ | |||
+ | ==Group properties== | ||
+ | |||
+ | {| class="wikitable" border="1" | ||
+ | !Property !! Satisfied !! Explanation | ||
+ | |- | ||
+ | |[[Abelian group]] || False for <math>n \ge 3</math> || For <math>n \ge 3</math>, the elements <math>a,x</math> do not commute. | ||
+ | |- | ||
+ | |[[Nilpotent group]] || True for <math>n</math> a power of <math>2</math>, false otherwise || | ||
+ | |- | ||
+ | |[[Metacyclic group]] || True || | ||
+ | |- | ||
+ | |[[Supersolvable group]] || True || | ||
+ | |- | ||
+ | |[[Solvable group]] || True || | ||
+ | |- | ||
+ | |[[T-group]] || True for <math>n</math> odd or twice an odd number, false for <math>n</math> a multiple of <math>4</math> || | ||
+ | |- | ||
+ | |[[Rational group]] || True for <math>n \le 4</math>, false for <math>n \ge 5</math> || | ||
+ | |- | ||
+ | |[[Rational-representation group]] || True for <math>n \le 4</math>, false for <math>n \ge 5</math> || | ||
+ | |- | ||
+ | |[[Ambivalent group]] || True || Elements in the cyclic subgroup are conjugate via <math>x</math>, elements outside have order two. | ||
+ | |} | ||
+ | |||
===In the case of twice an odd number=== | ===In the case of twice an odd number=== | ||
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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. | ||
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==Internal links== | ==Internal links== |
Revision as of 22:53, 26 August 2009
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
Contents
Definition
The dihedral group of degree and order , denoted sometimes as , sometimes as , and sometimes as (this wiki uses ) is defined in the following equivalent ways:
- It has the presentation:
- (For ): It is the group of symmetries of a regular -gon in the plane, viz., the plane isometries that preserves the set of points of the regular -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 and , 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 case of the dihedral group and is denoted and is defined as:
.
Related notions
Generalizations
- 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.
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 for some . It is abelian only if it has order or .
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
here denotes the degree, or half the order, of the dihedral group, which we denote as .
Function | Value | Explanation | |
---|---|---|---|
order | Cyclic subgroup of order , quotient of order . | ||
exponent | Exponent of cyclic subgroup is , elements outside have order . | ||
derived length | for , for | ||
nilpotency class | when , none otherwise | Nilpotent only if is a power of . | |
max-length | 1 + sum of exponents of prime divisors of | The dihedral groups are solvable. | |
composition length | 1 + sum of exponents of prime divisors of | The dihedral groups are solvable. | |
number of subgroups | is the divisor sum function and is the divisor count function. | ||
number of conjugacy classes | if is odd, if is even. |
Group properties
Group properties
Property | Satisfied | Explanation |
---|---|---|
Abelian group | False for | For , the elements do not commute. |
Nilpotent group | True for a power of , false otherwise | |
Metacyclic group | True | |
Supersolvable group | True | |
Solvable group | True | |
T-group | True for odd or twice an odd number, false for a multiple of | |
Rational group | True for , false for | |
Rational-representation group | True for , false for | |
Ambivalent group | True | Elements in the cyclic subgroup are conjugate via , elements outside have order two. |
In the case of twice an odd number
Suppose is odd and greater than . Let be the dihedral group of order . It has the following elements:
- elements of odd order, all of them in the cyclic subgroup of order . Of these, there are precisely elements of order for any divisor of .
- elements of order . These are the elements outside the cyclic subgroup .
The conjugacy classes of elements are as follows (a total of conjugacy classes):
- The identity element is its own conjugacy class.
- The non-identity elements in occur in conjugacy classes of size two. Each element is conjugate in to its inverse. Thus, these form conjugacy classes.
- The elements outside all form a single conjugacy class of size .
The equivalence classes of elements upto automorphism are as follows:
- Two elements inside are related via an automorphism if and only if they generate the same cyclic subgroup.
- Any two elements outside are related via an automorphism (in fact, they are in the same conjugacy class).
In the case of twice an even number
Suppose , and is the dihedral group of order . Then, has the following conjugacy classes (a total of conjugacy classes):
- There are conjugacy classes inside : The identity element and are both central elements. All other elements have a conjugacy class of size two: the element and its inverse.
- There are two conjugacy classes outside , of size each.
Subgroups
Further information: Subgroup structure of dihedral groups
There are two kinds of subgroups:
- Subgroups of the form , where . The number of such subgroups equals the number of positive divisors of , sometimes denoted . The subgroup generated by is a cyclic group of order .
- Subgroups of the form , where and . The number of such subgroups equals the sum of all positive divisors of , sometimes denoted . The subgroup of the above form is a dihedral group of order .
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
Supergroups
Groups having the dihedral group as quotient
The dicyclic group, also called the binary dihedral group, of order , has the dihedral group of order 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:
.
Dicyclic groups whose order is a power of 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 978-0471433347^{More 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)