construct a permutation group isomorphic to a projective special linear group
a positive integer
power of a prime number
The projective special linear group PSL⁡n,q is the quotient of the special linear group SL⁡n,q by its center.
The ProjectiveSpecialLinearGroup( n, q ) command returns a permutation group isomorphic to the projective special linear group PSL⁡n,q for the implemented ranges of the parameters n and q.
The implemented ranges for n and q are as follows:
n = 6,7,8,9,10
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.
with( GroupTheory ):
ProjectiveSpecialLinearGroup( 3, 2 );
GroupOrder( PSL( 3, 3 ) );
Note that PSL( 3, 4 ) has the same order as the alternating group of degree 8.
G := PSL( 3, 4 ):
GroupOrder( G );
GroupOrder( Alt( 8 ) );
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]] );
PermOrder( p );
Next, there is no element of order 15 in PSL( 3, 4 ).
ormap( g -> PermOrder( g ) = 15, Elements( G ) );
This shows that there are two non-isomorphic simple groups of order 20160.
IsSimple( G );
IsSimple( Alt( 8 ) );
Several among the small projective special linear groups are isomorphic to alternating groups.
AreIsomorphic( PSL( 2, 3 ), Alt( 4 ) );
AreIsomorphic( PSL( 2, 4 ), Alt( 5 ) );
AreIsomorphic( PSL( 2, 5 ), Alt( 5 ) );
AreIsomorphic( PSL( 2, 9 ), Alt( 6 ) );
GroupOrder( PSL( 4, q ) );
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.
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