WOLFRAM|DEMONSTRATIONS PROJECT

GL(2,p) and GL(3,3) Acting on Points

group

GL(2,3)

GL(2,5)

GL(2,7)

GL(2,11)

GL(3,3)

matrix

7

The two-dimensional space



3

×



3

contains nine points: (0,0), (0,1), (0,2), (1,0), (1,1), (1,2), (2,0), (2,1), and (2,2). The 48 invertible 2×2 matrices over



3

form the general linear group known as

GL(2,3)

. They act on



3

×



3

by matrix multiplication modulo 3, permuting the nine points. More generally,

GL(n,p)

is the set of invertible

n×n

matrices over the field



p

, where

p

is prime

.

With (0,0) shifted to the center, the matrix actions on the nine points make symmetrical patterns.