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GroupTheory

  

ProjectiveSpecialLinearGroup

  

construct a permutation group isomorphic to a projective special linear group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

ProjectiveSpecialLinearGroup(n, q)

PSL(n, q)

Parameters

n

-

a positive integer

q

-

power of a prime number

Description

• 

The projective special linear group PSLn,q is the quotient of the special linear group SLn,q by its center.

• 

The ProjectiveSpecialLinearGroup( n, q ) command returns a permutation group isomorphic to the projective special linear group PSLn,q for the implemented ranges of the parameters n and q.

• 

The implemented ranges for n and q are as follows:

n=2

q241

n=3

q20

n=4

q10

n=5

q5

n = 6,7,8,9,10

q=2

• 

The command PSL( n, q ) is provided as an abbreviation.

Examples

withGroupTheory:

ProjectiveSpecialLinearGroup3,2

PSL3,2

(1)

GroupOrderPSL3,3

5616

(2)

Note that PSL( 3, 4 ) has the same order as the alternating group of degree 8.

GPSL3,4:

GroupOrderG

20160

(3)

GroupOrderAlt8

20160

(4)

However, PSL( 3, 4 ) and Alt( 8 ) are not isomorphic.  First, Alt( 8 ) has an element of order equal to 15.

pPerm1,2,3,4,5,6,7,8

p:=1,2,3,4,56,7,8

(5)

PermOrderp

15

(6)

Next, there is no element of order 15 in PSL( 3, 4 ).

ormapg→PermOrderg=15,ElementsG

false

(7)

This shows that there are two non-isomorphic simple groups of order 20160.

IsSimpleG

true

(8)

IsSimpleAlt8

true

(9)

AreIsomorphicPSL2,3,Alt4

true

(10)

AreIsomorphicPSL2,4,Alt5

true

(11)

AreIsomorphicPSL2,5,Alt5

true

(12)

AreIsomorphicPSL2,9,Alt6

true

(13)

Compatibility

• 

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

• 

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

See Also

GroupTheory[GroupOrder]

GroupTheory[IsSimple]

GroupTheory[ProjectiveSpecialUnitaryGroup]

GroupTheory[SpecialLinearGroup]

 


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