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GroupTheory

  

SpecialOrthogonalGroup

  

construct a permutation group isomorphic to a special orthogonal group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

SpecialOrthogonalGroup(d, n, q)

Parameters

d

-

0, 1 or -1

n

-

a positive integer

q

-

power of a prime number

Description

• 

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 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

q100

n=3

q20

n=4

q10

n=5

q5

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.

Examples

withGroupTheory:

SpecialOrthogonalGroup0,9,2

GroupTheory:-GeneralOrthogonalGroup0,9,2

(1)

GSpecialOrthogonalGroup1,4,7

GroupTheory:-SpecialOrthogonalGroup1,4,7

(2)

DegreeG

128

(3)

GroupOrderG

112896

(4)

IsTransitiveG

true

(5)

GSpecialOrthogonalGroup1,4,7

GroupTheory:-SpecialOrthogonalGroup1,4,7

(6)

DegreeG

100

(7)

GroupOrderG

117600

(8)

IsTransitiveG

true

(9)

GroupOrderSpecialOrthogonalGroup0,7,3

9170703360

(10)

IsSimpleDerivedSubgroupSpecialOrthogonalGroup1,4,8

true

(11)

IsSimpleDerivedSubgroupSpecialOrthogonalGroup1,4,8

false

(12)

GSpecialOrthogonalGroup0,5,q

GroupTheory:-SpecialOrthogonalGroup0,5,q

(13)

GroupOrderG

q4q21q41

(14)

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

1

18

72

8

8

32

32

6

6

36

36

8

8

32

32

72

72

48

48

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

χ__1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

χ__2

1

1

1

−1

1

1

1

1

1

1

−1

−1

1

1

1

1

−1

−1

1

1

χ__3

2

2

2

0

−1

−1

−1

2

2

2

0

0

−1

−1

−1

2

0

0

−1

−1

χ__4

2

2

2

0

−1

−1

2

−1

2

2

0

0

−1

−1

2

−1

0

0

−1

−1

χ__5

2

2

2

0

−1

2

−1

−1

2

2

0

0

2

−1

−1

−1

0

0

−1

2

χ__6

2

2

2

0

2

−1

−1

−1

2

2

0

0

−1

2

−1

−1

0

0

2

−1

χ__7

3

3

−1

−1

3

0

0

0

3

−1

1

1

0

3

0

0

1

−1

−1

0

χ__8

3

3

−1

1

3

0

0

0

3

−1

−1

−1

0

3

0

0

−1

1

−1

0

χ__9

3

3

−1

−1

0

3

0

0

−1

3

1

1

3

0

0

0

−1

1

0

−1

χ__10

3

3

−1

1

0

3

0

0

−1

3

−1

−1

3

0

0

0

1

−1

0

−1

χ__11

4

−4

0

0

−2

−2

1

1

0

0

−2

2

2

2

−1

−1

0

0

0

0

χ__12

4

−4

0

0

−2

−2

1

1

0

0

2

−2

2

2

−1

−1

0

0

0

0

χ__13

6

6

−2

0

0

−3

0

0

−2

6

0

0

−3

0

0

0

0

0

0

1

χ__14

6

6

−2

0

−3

0

0

0

6

−2

0

0

0

−3

0

0

0

0

1

0

χ__15

8

−8

0

0

2

2

−1

2

0

0

0

0

−2

−2

1

−2

0

0