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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.

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