QuasiDihedralGroup - Maple Help
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GroupTheory

  

SemiDihedralGroup

  

construct a semi-dihedral group as a permutation group or a finitely presented group

  

QuasiDihedralGroup

  

construct a quasi-dihedral group as a permutation group or a finitely presented group

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

SemiDihedralGroup( n, formopt )

QuasiDihedralGroup( n, formopt )

Parameters

n

-

algebraic; understood to be an integer greater than

formopt

-

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

Description

• 

The semi-dihedral of degree  is a non-abelian group of order  which contains a cyclic subgroup of order  for . It is defined by a presentation of the form

• 

The SemiDihedralGroup( n ) command returns a semi-dihedral group, either as a permutation group (the default) or as a finitely presented group.

• 

You can specify the form of the group returned explicitly by passing one of the options 'form' = "permgroup" or 'form' = "fpgroup".

• 

If the parameter n is not a positive integer, then a symbolic group representing the semi-dihedral group of order 8*n is returned.

• 

If  is a power of , the resulting group is a quasi-dihedral group. In other words, a quasi-dihedral group is a semi-dihedral -group. (This is analogous to the fact that a quaternion group is a dicyclic -group.) A semi-dihedral group is nilpotent only if it is quasi-dihedral.

• 

The QuasiDihedralGroup( n ) command returns a quasi-dihedral group of order , provided that n is an integer greater than . If n is a non-numeric algebraic expression, then a symbolic group representing the quasi-dihedral group of order  is returned.

Examples

(1)

(2)

(3)

(4)

(5)

(6)

(7)

The center of a semi-dihedral group is always cyclic, but the order depends upon whether  is odd or even. For odd , the center has order .

(8)

(9)

For even , the center has order .

(10)

(11)

The permutation representation used in Maple is always transitive, but imprimitive.

(12)

(13)

(14)

(15)

(16)

Use the form = "fpgroup" option to construct a finitely presented semi-dihedral group.

(17)

(18)

Note that dihedral and semi-dihedral groups of the same order are non-isomorphic.

(19)

C

1a

2a

2b

2c

4a

4b

4c

4d

5a

5b

10a

10b

20a

20b

20c

20d

|C|

1

1

5

5

1

1

5

5

2

2

2

2

2

2

2

2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(20)

(21)

(22)

(23)

(24)

(25)

(26)

See Also

GroupTheory[CharacterTable]

GroupTheory[ClassNumber]

GroupTheory[GroupOrder]

 


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