# GAP:GeneralLinearGroup

## Definition

### Function type

This function takes as input two arguments, the first of which is a positive integer, and the second is either a ring or a prime power.

The function can be called using either GeneralLinearGroup and GL.

### Behavior

• If the first argument is a positive integer $n$ and the second argument is a prime power $q$, the argument returns the general linear group of degree $n$ (i.e., of $n \times n$ matrices) over the field of $q$ elements.
• If the first argument is a positive integer $n$ and the second argument is a ring, the argument returns the general linear group of degree $n$ over that ring.
• If the second argument is an integer that is not a prime power, GAP returns a usage error.
• If the first argument is an integer that is not positive, GAP returns a NoMethodFound error.

## Examples of usage

Here is an example:

gap> G := GL(2,3);
GL(2,3)
gap> IdGroup(G);
[ 48, 29 ]
gap> IsSolvable(G);
true

Here is another example:

gap> G := GeneralLinearGroup(3,5);
GL(3,5)
gap> C := Center(G);
<group of 3x3 matrices in characteristic 5>
gap> H := G/C;
Group([ (8,11,10,9)(12,27,22,17)(13,31,25,19)(14,28,26,20)(15,29,23,21)(16,30,24,18), (1,2,7,10,25,3,12,9,20,31,18,21,16,24,4,17,11,30,13,29,28,23,5,
22)(6,27,8,15,19,26) ])