GroupTheory
GeneralUnitaryGroup
construct a permutation group isomorphic to a general unitary group
Calling Sequence
Parameters
Description
Examples
Compatibility
GeneralUnitaryGroup(n, q)
n
-
a positive integer
q
power of a prime number
The general unitary group GU⁡n,q (often denoted by U⁡n,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 GU⁡n,q for the implemented ranges of the parameters n and q.
The implemented ranges for n and q are as follows:
n=2
q≤20
n=3
q≤5
n=4
q≤4
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.
with⁡GroupTheory:
GeneralUnitaryGroup⁡2,2
GU2,2
GroupOrder⁡GeneralUnitaryGroup⁡2,4
300
IdentifySmallGroup⁡GeneralUnitaryGroup⁡2,4
300,22
GroupOrder⁡GeneralUnitaryGroup⁡4,q
q+1⁢q6⁢q2−1⁢q3+1⁢q4−1
simplify⁡GroupOrder⁡GeneralUnitaryGroup⁡2,3k
81k+27k−9k−3k
simplify⁡ClassNumber⁡GeneralUnitaryGroup⁡2,3k
9k+2⁢3k+1
Here is a general formula for the order of the general unitary group of dimension n over a field of order q.
GroupOrder⁡GeneralUnitaryGroup⁡n,q
q+1⁢qn⁢n−12⁢∏k=1n−1⁡qk+1−−1k+1
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]
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