GroupTheory
SpecialOrthogonalGroup
construct a permutation group isomorphic to a special orthogonal group
Calling Sequence
Parameters
Description
Examples
Compatibility
SpecialOrthogonalGroup(d, n, q)
d
-
0, 1 or -1
n
a positive integer
q
power of a prime number
The special orthogonal group SO⁡d,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 SO⁡d,n,q and GO⁡d,n,q are isomorphic.
The SpecialOrthogonalGroup( d, n, q ) command returns a permutation group isomorphic to the special orthogonal group SO⁡d,n,q for values of the parameters d, n and q in the implemented ranges.
The implemented ranges for n and q are as follows:
n=2
q≤100
n=3
q≤20
n=4
q≤10
n=5
q≤5
n=6,7,8
q=3
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.)
In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.
with⁡GroupTheory:
SpecialOrthogonalGroup⁡0,9,2
GroupTheory:-GeneralOrthogonalGroup⁡0,9,2
G≔SpecialOrthogonalGroup⁡1,4,7
GroupTheory:-SpecialOrthogonalGroup⁡1,4,7
Degree⁡G
128
GroupOrder⁡G
112896
IsTransitive⁡G
true
G≔SpecialOrthogonalGroup⁡−1,4,7
GroupTheory:-SpecialOrthogonalGroup⁡−1,4,7
100
117600
GroupOrder⁡SpecialOrthogonalGroup⁡0,7,3
9170703360
IsSimple⁡DerivedSubgroup⁡SpecialOrthogonalGroup⁡−1,4,8
IsSimple⁡DerivedSubgroup⁡SpecialOrthogonalGroup⁡1,4,8
false
G≔SpecialOrthogonalGroup⁡0,5,q
GroupTheory:-SpecialOrthogonalGroup⁡0,5,q
q4⁢q2−1⁢q4−1
Display⁡CharacterTable⁡SpecialOrthogonalGroup⁡1,4,3
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