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GroupTheory

  

GeneralUnitaryGroup

  

construct a permutation group isomorphic to a general unitary group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

GeneralUnitaryGroup(n, q)

Parameters

n

-

a positive integer

q

-

power of a prime number

Description

• 

The general unitary group GUn,q (often denoted by Un,q) is the group of all n×n matrices over the field with q2 elements, where q is a prime power, that respect a fixed nondegenerate sesquilinear form.

• 

The GeneralUnitaryGroup( n, q ) command returns a permutation group isomorphic to the general unitary group GUn,q  for the implemented ranges of the parameters n and q.

• 

The implemented ranges for n and q are as follows:

n=2

q20

n=3

q5

n=4

q4

n=5,6

q=2

• 

If either, or both, of n and q is non-numeric, then a symbolic group representing the general unitary group 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:

GeneralUnitaryGroup2,2

GU2,2

(1)

GroupOrderGeneralUnitaryGroup2,4

300

(2)

IdentifySmallGroupGeneralUnitaryGroup2,4

300,22

(3)

GroupOrderGeneralUnitaryGroup4,q

q+1q6q21q3+1q41

(4)

simplifyGroupOrderGeneralUnitaryGroup2,3k

81k+27k9k3k

(5)

simplifyClassNumberGeneralUnitaryGroup2,3k

9k+23k+1

(6)

Here is a general formula for the order of the general unitary group of dimension n over a field of order q.

GroupOrderGeneralUnitaryGroupn,q

q+1qnn12k=1n1qk+1−1k+1

(7)

Compatibility

• 

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

• 

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

• 

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

See Also

GroupTheory[GeneralLinearGroup]

GroupTheory[GroupOrder]

GroupTheory[ProjectiveGeneralUnitaryGroup]