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GroupTheory

  

ProjectiveGeneralLinearGroup

  

construct a permutation group isomorphic to a projective general linear group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

ProjectiveGeneralLinearGroup(n, q)

PGL(n, q)

Parameters

n

-

a positive integer

q

-

power of a prime number

Description

• 

The projective general linear group PGLn,q  is the quotient of the general linear group GLn,q  by its center.

• 

The ProjectiveGeneralLinearGroup( n, q ) command returns a permutation group isomorphic to the projective general linear group PGLn,q  . The PGL(n, q) command is provided as an alias.

• 

The ranges for n and q are as follows:

n=2

q100

n=3

q20

n=4

q10

n=5

q5

n=6,7,8,9,10

q=2

• 

If either of n or q is non-numeric, then a symbolic group 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:

GProjectiveGeneralLinearGroup2,49

GPGL2,49

(1)

GeneratorsG

3,50,49,48,47,46,45,44,8,43,42,41,40,39,38,37,7,36,35,34,33,32,31,30,6,29,28,27,26,25,24,23,5,22,21,20,19,18,17,16,4,15,14,13,12,11,10,9,1,2,63,4,89,25,3310,20,3711,32,2412,15,4113,38,1614,30,2317,31,1918,44,4721,46,4322,39,4926,34,5027,48,3528,42,4029,45,36

(2)

GProjectiveGeneralLinearGroup5,2

GPGL5,2

(3)

GroupOrderG

9999360

(4)

GroupOrderPGL4,3

12130560

(5)

Several among the small projective general linear groups are isomorphic to some familiar permutation groups.

AreIsomorphicPGL2,2,Symm3

true

(6)

AreIsomorphicPGL2,3,Symm4

true

(7)

AreIsomorphicPGL2,4,Alt5

true

(8)

AreIsomorphicPGL2,5,Symm5

true

(9)

GroupOrderPGL3,q

q31q3qq3q2q1

(10)

Compatibility

• 

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

• 

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

• 

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

See Also

GroupTheory[AlternatingGroup]

GroupTheory[AreIsomorphic]

GroupTheory[GeneralLinearGroup]

GroupTheory[GroupOrder]

GroupTheory[SymmetricGroup]