DihedralGroup - Maple Help
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GroupTheory

  

DihedralGroup

  

construct a dihedral group of a given degree

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

DihedralGroup( n )

DihedralGroup( n, s )

Parameters

n

-

: algebraic : an expression understood to be a positive integer or

s

-

: equation : (optional) equation of the form form = "fpgroup" or form = "permgroup" (default)

Description

• 

The dihedral group of degree n is the symmetry group of an n-sided regular polygon for n>2. It is generated by a reflection (of order 2), and a rotation (of order n). It acts as a permutation group on the vertices of the regular n-sided polygon.

• 

For n=1, the dihedral group is a cyclic group of order 2.  For n=2, the dihedral group is the non-cyclic group of order 4, also known as the Klein 4-group.

• 

If n=, then an infinite dihedral group (a free product of two groups of order two, or the holomorph of an infinite cyclic group) is returned as a finitely presented group.

• 

The DihedralGroup( n ) command returns a dihedral group, either as a permutation group or a group defined by generators and defining relations. By default, if n is a positive integer, then a permutation group is returned, but a finitely presented group can be requested by passing the option 'form' = "fpgroup". If n= then a finitely presented group is returned, regardless of any form option passed.

• 

If the value of the parameter n is not numeric, then a symbolic group representing the dihedral group of the indicated degree is returned.

• 

In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

Examples

withGroupTheory:

GDihedralGroup13

GD13

(1)

GroupOrderG

26

(2)

GDihedralGroup13,form=fpgroup

GD13

(3)

GDihedralGroup17,form=permgroup

GD17

(4)

GroupOrderG

34

(5)

AreIsomorphicDihedralGroup3,Symm3

true

(6)

GroupOrderDihedralGroup3k

6k

(7)

IsNilpotentDihedralGroup6kassumingk::posint

false

(8)

IsNilpotentDihedralGroup2a4bassumingposint

true

(9)

IsFrobeniusGroupDihedralGroup7

true

(10)

IsFrobeniusGroupDihedralGroup6

false

(11)

DrawCayleyTableDihedralGroup5,conjugacy=true

ClassNumberDihedralGroup6nassumingn::posint

3n+3

(12)

ExponentDihedralGroup2n+1assumingn::posint

4n+2

(13)

IsPerfectOrderClassesGroupDihedralGroup9

true

(14)

IsPerfectOrderClassesGroupDihedralGroup10

false

(15)

GDihedralGroup

GD

(16)

IsNilpotentG

false

(17)

IsSupersolubleG

true

(18)

IdentifyFrobeniusGroupDihedralGroup11

22,1

(19)

DisplayCharacterTableDihedralGroup5

C

1a

2a

5a

5b

|C|

1

5

2

2

 

 

 

 

 

chi__1

1

1

1

1

chi__2

1

−1

1

1

chi__3

2

0

−125−135

−135−1251

chi__4

2

0

−135−1251

−125−135

caygrCayleyGraphDihedralGroup4

caygrGraph 1: a directed graph with 8 vertices and 16 arcs

(20)

GraphTheory:-DrawGraphcaygr

Compatibility

• 

The GroupTheory[DihedralGroup] command was introduced in Maple 17.

• 

For more information on Maple 17 changes, see Updates in Maple 17.

See Also

GroupTheory[DicyclicGroup]

GroupTheory[GroupOrder]

GroupTheory[IsNilpotent]

 


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