GroupTheory
SpecialOrthogonalGroup
construct a permutation group isomorphic to a special orthogonal group
Calling Sequence
Parameters
Description
Examples
Compatibility
SpecialOrthogonalGroup(d, n, q)
SO(d, n, q)
d
-
0, 1 or -1
n
a positive integer
q
power of a prime number
The special orthogonal group SOd,n,q is the set of all n×n matrices over the field with q elements that respect a non-singular quadratic form and have determinant equal to 1. The value of d must be 0 for odd values of n, or 1 or −1 for even values of n. Note that for even values of q the groups SOd,n,q and GOd,n,q are isomorphic.
The SpecialOrthogonalGroup( d, n, q ) command returns a permutation group isomorphic to the special orthogonal group SOd,n,q .
If either or both of the parameters n and q is non-numeric, then a symbolic group representing the indicated special orthogonal group is returned. (The argument d must be numeric, equal to one of 0, 1 or −1.)
The command SO(d, n, q) is provided as an alias.
In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.
withGroupTheory:
SpecialOrthogonalGroup0,9,2
GO0,9,2
G≔SpecialOrthogonalGroup1,4,7
G≔SO1,4,7
DegreeG
128
GroupOrderG
112896
IsTransitiveG
true
G≔SpecialOrthogonalGroup−1,4,7
G≔SO-1,4,7
100
117600
GroupOrderSpecialOrthogonalGroup0,7,3
9170703360
IsSimpleDerivedSubgroupSpecialOrthogonalGroup−1,4,8
IsSimpleDerivedSubgroupSpecialOrthogonalGroup1,4,8
false
G≔SpecialOrthogonalGroup0,5,q
G≔SO0,5,q
q4q2−1q4−1
DisplayCharacterTableSpecialOrthogonalGroup1,4,3
C
1a
2a
2b
2c
3a
3b
3c
3d
4a
4b
4c
4d
6a
6b
6c
6d
8a
8b
12a
12b
|C|
1
18
72
8
32
6
36
48
χ__1
χ__2
−1
χ__3
2
0
χ__4
χ__5
χ__6
χ__7
3
χ__8
χ__9
χ__10
χ__11
4
−4
−2
χ__12
χ__13
−3
χ__14
χ__15
−8
χ__16
χ__17
χ__18
χ__19
9
χ__20
The GroupTheory[SpecialOrthogonalGroup] command was introduced in Maple 17.
For more information on Maple 17 changes, see Updates in Maple 17.
The GroupTheory[SpecialOrthogonalGroup] command was updated in Maple 2020.
See Also
GroupTheory[Degree]
GroupTheory[GeneralOrthogonalGroup]
GroupTheory[GroupOrder]
GroupTheory[IsTransitive]
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