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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

• 

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

with( GroupTheory ):

ProjectiveSpecialLinearGroup( 3, 2 );

PSL3,2

(1)

GroupOrder( PSL( 3, 3 ) );

5616

(2)

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

G := PSL( 3, 4 ):

GroupOrder( G );

20160

(3)

GroupOrder( Alt( 8 ) );

20160

(4)

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

p := Perm( [[1,2,3,4,5],[6,7,8]] );

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

(5)

PermOrder( p );

15

(6)

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

ormap( g -> PermOrder( g ) = 15, Elements( G ) );

false

(7)

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

IsSimple( G );

true

(8)

IsSimple( Alt( 8 ) );

true

(9)

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

AreIsomorphic( PSL( 2, 3 ), Alt( 4 ) );

true

(10)

AreIsomorphic( PSL( 2, 4 ), Alt( 5 ) );

true

(11)

AreIsomorphic( PSL( 2, 5 ), Alt( 5 ) );

true

(12)

AreIsomorphic( PSL( 2, 9 ), Alt( 6 ) );

true

(13)

GroupOrder( PSL( 4, 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]