GAP:SpecialLinearGroup

This article is about a GAP function.

Definition

Function type

The function takes as input two arguments, one of which is a positive integer and the other is either a prime power or a ring.

The function can be written as SpecialLinearGroup or SL.

Behavior

The function outputs a group stored in the matrix group format over the appropriate field of ring (the storage format can be verified using the IsMatrixGroup function).

• If the first argument is a positive integer $n$ and the second argument is a prime power $q$, the function returns the special linear group of degree $n$ over the field of $q$ elements.
• If the first argument is a positive integer $n$ and the second argument is a ring, the function returns the special linear group of degree $n$ over the ring.

Related functions

• GAP:GeneralLinearGroup
• GAP:ProjectiveGeneralLinearGroup
• GAP:ProjectiveSpecialLinearGroup

Memory usage

For the function as a whole

The memory usage for the function is 1540.

For particular groups constructed using the function

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Examples of usage

Here is an example that also uses GAP:GeneralLinearGroup.

gap> G := GL(2,3);
GL(2,3)
gap> H := SL(2,3);
SL(2,3)
gap> IsMatrixGroup(H);
true
gap> IsSubgroup(G,H);
true
gap> IsNormal(G,H);
true

gap> G := GeneralLinearGroup(2,2);
SL(2,2)
gap> H := SpecialLinearGroup(2,2);
SL(2,2)
gap> G = H;
true