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GroupTheory

  

GeneralLinearGroup

  

construct a permutation group isomorphic to the General Linear Group over a finite field

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

GeneralLinearGroup(n, q)

GL( n, q )

Parameters

n

-

a positive integer

q

-

power of a prime number

Description

• 

The general linear group GLn,q is the set of all nonsingular n×n matrices over a finite field of size q, where q is a prime power.

• 

If n and q are positive integers, then the GeneralLinearGroup( n, q ) command returns a permutation group isomorphic to the general linear group  GLn,q for the implemented ranges of the parameters n and q. Otherwise, a symbolic group is returned, for which Maple can do some limited computations.

• 

The implemented ranges for n and q are as follows:

n=2

q100

n=3

q20

n=4

q10

n=5

q5

n=6,7,8,9,10

q=2

• 

The abbreviation GL( n, q ) is available as a synonym for GeneralLinearGroup( n, q ).

• 

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

Examples

withGroupTheory:

GeneralLinearGroup2,3

GL2,3

(1)

GGL2,5

GGL2,5

(2)

GroupOrderG

480

(3)

csCompositionSeriesG

csGL2&comma;5 < a permutation group on 24 letters with 5 generators > 1&comma;32&comma;45&comma;156&comma;187&comma;198&comma;169&comma;1710&comma;2011&comma;2312&comma;2413&comma;2114&comma;22

(4)

seqGroupOrderS&comma;S=cs

480,240,120,2,1

(5)

GroupOrderGL4&comma;3

24261120

(6)

ClassNumberGL31&comma;q

q31q15q14q13q12q11q10+2q8+3q7+4q6+q53q43q3+q

(7)

GroupOrderGLn&comma;q

k=0n1qnqk

(8)

GroupOrderGL3&comma;q

q31q3qq3q2

(9)

GroupOrderDerivedSubgroupGLn&comma;q

qn2k=1n1qk+11

(10)

Compatibility

• 

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

• 

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

• 

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

See Also

GroupTheory[GeneralOrthogonalGroup]

GroupTheory[GeneralUnitaryGroup]

GroupTheory[GroupOrder]

GroupTheory[ProjectiveGeneralLinearGroup]