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GroupTheory

  

GeneralOrthogonalGroup

  

construct a permutation group isomorphic to a general orthogonal group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

GeneralOrthogonalGroup(d, n, q)

Parameters

d

-

0, 1 or -1

n

-

a positive integer

q

-

power of a prime number

Description

• 

The general orthogonal group GOd,n,q is the set of all n×n matrices over the field with q elements that respect a non-singular quadratic form. The value of d must be 0 for odd n, or 1 or −1 for even n.

• 

The GeneralOrthogonalGroup( d, n, q ) command returns a permutation group isomorphic to the general orthogonal group GOd,n,q for the implemented values of d, n and q.

• 

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

n=9,10,11

q=2

• 

If the argument q is not a prime power (and is non-numeric), then a symbolic group representing GOd,n,q  is returned.

• 

In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

Examples

withGroupTheory:

GGeneralOrthogonalGroup0,7,2

GGO7,2

(1)

GeneratorsG

5,67,98,1011,1413,1715,2018,1921,2622,2823,3024,3225,3429,3731,4033,4336,4638,4139,4942,4445,5248,5153,5857,6259,61,1,2,3,4,5,76,8,11,15,21,279,12,16,22,29,3810,13,18,23,31,4114,19,24,33,44,3417,20,2526,35,45,53,32,4228,36,4730,39,50,56,61,5837,4840,51,46,54,59,6249,55,60,63,52,57

(2)

GGeneralOrthogonalGroup1,4,5

GGO4,5

(3)

GroupOrderG

28800

(4)

GGeneralOrthogonalGroup1,4,5

GGO4,5

(5)

DegreeG

130

(6)

GroupOrderG

31200

(7)

GGeneralOrthogonalGroup0,3,5

GGO3,5

(8)

CharacterTableG

(9)

OrderClassPolynomialG,x

24x10+60x6+24x5+60x4+20x3+51x2+x

(10)

DerivedSeriesG

GO3,51,42,83,95,156,237,2010,1211,1913,1614,2417,2118,22,1,42,83,225,67,189,1011,2312,1713,1415,1619,2420,21,1,202,193,94,215,156,167,128,2410,2211,1314,2317,18

(11)

HypercentreG

1,42,83,95,156,167,1710,2211,1412,1813,2319,2420,21

(12)

IsMalnormalSylowSubgroup2,G,G

false

(13)

GroupOrderPCore2,G

2

(14)

IsMalnormalSylowSubgroup3,G,G

true

(15)

IsMalnormalSylowSubgroup5,G,G

true

(16)

GroupOrderGeneralOrthogonalGroup0,7,3

18341406720

(17)

GroupOrderGeneralOrthogonalGroup1,8,2

348364800

(18)

GroupOrderGeneralOrthogonalGroup1,8,2

394813440

(19)

GroupOrderGeneralOrthogonalGroup1,4,q

igcd2,q1igcd2,qq2q2+1q21

(20)

GroupOrderGeneralOrthogonalGroup1,4,q

igcd2,q1igcd2,qq2q212

(21)

Compatibility

• 

The GroupTheory[GeneralOrthogonalGroup] command was introduced in Maple 17.

• 

For more information on Maple 17 changes, see Updates in Maple 17.

• 

The GroupTheory[GeneralOrthogonalGroup] command was updated in Maple 2020.

See Also

GroupTheory[Degree]

GroupTheory[GeneralLinearGroup]

GroupTheory[GroupOrder]

GroupTheory[SpecialOrthogonalGroup]