GroupTheory - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Group Theory : GroupTheory/ProjectiveSpecialLinearGroup

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

• 

If either, or both, of n and q is non-numeric, then a symbolic group representing the symplectic group is returned.

• 

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

• 

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

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

p1,2,3,4,56,7,8

(5)

PermOrderp

15

(6)

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

ormapgPermOrderg=15,ElementsG

false

(7)

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

IsSimpleG

true

(8)

IsSimpleAlt8

true

(9)

Several among the small projective special linear groups are isomorphic to alternating groups.

AreIsomorphicPSL2,3,Alt4

true

(10)

AreIsomorphicPSL2,4,Alt5

true

(11)

AreIsomorphicPSL2,5,Alt5

true

(12)

AreIsomorphicPSL2,9,Alt6

true

(13)

GroupOrderPSL4,q

q6q21q31q41igcd4,q1

(14)

Compatibility

• 

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

• 

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

• 

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

See Also

GroupTheory[GroupOrder]

GroupTheory[IsSimple]

GroupTheory[ProjectiveSpecialUnitaryGroup]

GroupTheory[SpecialLinearGroup]