GroupTheory
SymplecticGroup
construct a permutation group isomorphic to a symplectic group
Calling Sequence
Parameters
Description
Examples
Compatibility
SymplecticGroup(n, q)
Sp(n, q)
n
-
an even positive integer
q
power of a prime number
The symplectic group Spn,q is the group of all n×n matrices over the field with q elements that respect a fixed nondegenerate symplectic form. The integer n must be even.
The SymplecticGroup( n, q ) command returns a permutation group isomorphic to the symplectic group Spn,q .
Note that for n=2 the groups Spn,q and SLn,q are isomorphic, so that a special linear group is returned in this case.
If either, or both, of n and q is non-numeric, then a symbolic group representing the symplectic group is returned.
The Sp( n, q ) command 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.
withGroupTheory:
G≔SymplecticGroup4,5
G≔Sp4,5
ifactorGroupOrderG
27325413
GroupOrderSylowSubgroup2,G
128
S3≔SylowSubgroup3,G
S3≔⟨a permutation group on 624 letters with 2 generators⟩
GroupOrderS3
9
IsCyclicS3
false
IdentifySmallGroupS3
9,2
GroupOrderSylowSubgroup5,G
625
IsTrivialPCore5,G
true
GroupOrderSylowSubgroup13,G
13
G≔SymplecticGroup4,3
G≔Sp4,3
DegreeG
80
IsSimpleG
GroupOrderCentreG
2
For n=2 the corresponding special linear group is returned.
SymplecticGroup2,5
SL2,5
Note the exceptional isomorphism:
AreIsomorphicSymplecticGroup4,2,Symm6
G≔SymplecticGroup6,q
G≔Sp6,q
GroupOrderG
q9q2−1q4−1q6−1
ClassNumberSymplecticGroup8,q
5q+q+1q+4q2+q2+q+3q+q4+q3+7q::even25q+51+q+4q+11q2+q2+4q+10q+q4+4q3otherwise
ClassNumberSymplecticGroup4,11kassumingk::posint
511k+10+11k2
The GroupTheory[SymplecticGroup] command was introduced in Maple 17.
For more information on Maple 17 changes, see Updates in Maple 17.
The GroupTheory[SymplecticGroup] command was updated in Maple 2020.
See Also
GroupTheory[AreIsomorphic]
GroupTheory[ClassNumber]
GroupTheory[Degree]
GroupTheory[Generators]
GroupTheory[GroupOrder]
GroupTheory[ProjectiveSymplecticGroup]
GroupTheory[SpecialLinearGroup]
GroupTheory[SymmetricGroup]
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