Difference between revisions of "Dihedral group"
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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> | + | * 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, 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 has order <math>2n/ | + | * 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>. |
+ | |||
+ | 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]]}} | ||
+ | |||
+ | ==Supergroups== | ||
+ | |||
+ | ===Groups having the dihedral group as quotient=== | ||
+ | |||
+ | The [[dicyclic group]], also called the binary dihedral group, of order <math>4n</math>, has the dihedral group of order <math>2n</math> 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: | ||
+ | |||
+ | <math>\langle a,x \mid a^n = x^2 = (ax)^2 \}</math>. | ||
+ | |||
+ | Dicyclic groups whose order is a power of <math>2</math> are termed [[generalized quaternion group]]s. | ||
==Particular cases== | ==Particular cases== |
Revision as of 20:39, 16 March 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
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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 with parameter , denoted sometimes as and sometimes as 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.
Note that for and , the geometric description of the dihedral group does not make sense. In these cases, we use the algebraic description.
Elements
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
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.
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 | 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 |
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 |
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)