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GroupTheory

  

ProjectiveSpecialUnitaryGroup

  

construct a permutation group isomorphic to a projective special unitary group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

ProjectiveSpecialUnitaryGroup( n, q )

PSU( n, q )

Parameters

n

-

a positive integer

q

-

power of a prime number

Description

• 

The projective special unitary group PSUn,q  , over the field with q2 elements, is the quotient of the special unitary group SUn,q  by its center.

• 

Note that for n=2 the groups PSUn,q  andPSLn,q  are isomorphic. These groups are soluble being isomorphic, respectively, to the symmetric group of order 6, and the alternating group of order 12. Furthermore, the group PSU3,2  is a Frobenius group of order 72 and is soluble. For all other values of n and q, the groupPSUn,q  is simple.

• 

The ProjectiveSpecialUnitaryGroup( n, q ) command returns a permutation group isomorphic to the projective special unitary group PSUn,q  for values of the parameters n and q in the implemented ranges.

• 

The implemented ranges for n and q are as follows:

n=2

q241

n=3

q16

n=4

q5

n=5

q4

n=6

q3

n=7,8

q=2

• 

If either or both of the arguments n and q are non-numeric, then a symbolic group representing the projective special unitary group is returned.

• 

The command PSU( n, q ) is provided as an abbreviation.

• 

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

Examples

withGroupTheory:

GProjectiveSpecialUnitaryGroup3,3

GPSU3,3

(1)

DegreeG

28

(2)

GeneratorsG

3,4,6,10,12,18,19,235,8,13,20,17,11,16,79,14,21,15,22,24,26,2825,27,1,2,3,5,9,15,16,184,7,12,19,24,27,26,236,11,17,13,8,10,14,2120,25

(3)

IsSolublePSU2,2

true

(4)

AreIsomorphicPSU2,3,Alt4

true

(5)

IdentifyFrobeniusGroupPSU3,2

72,2

(6)

GroupOrderPSU5,3

258190571520

(7)

GroupOrderPSU4,q

q6q21q3+1q41igcd4,q+1

(8)

IsSimplePSU5,q

true

(9)

IsSimplePSU3&comma;qassuming3<q

true

(10)

IsTransitivePSU6&comma;2

false

(11)

orbsOrbitsPSU6&comma;2&colon;

mapnumelems&comma;orbs

693&comma;672

(12)

Compatibility

• 

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

• 

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

• 

The GroupTheory[ProjectiveSpecialUnitaryGroup] command was updated in Maple 2020.

See Also

assuming

GroupTheory[AreIsomorphic]

GroupTheory[Degree]

GroupTheory[Generators]

GroupTheory[GroupOrder]

GroupTheory[IdentifyFrobeniusGroup]

GroupTheory[IsSimple]

GroupTheory[ProjectiveSpecialLinearGroup]

GroupTheory[SpecialUnitaryGroup]