"The group GL(2,Z) can be identified with the group of continuous automorphisms of the torus."Historically, of course, visualization of GL(2,Z) preceded that of GL(2,p). See, for instance, the Wikipedia article on Arnold's Cat Map and especially its illustration (animated gif, 3.5 mb) of a discrete version of an iterated toral automorphism applied to a picture of a pepper.
|Steven H. Cullinane
Visualizing GL(2,p). Expository Note. March 26, 1985
"The typical example of a finite group is GL(n,q), the general linear group of n dimensions over the field with q elements." -- J. L. Alperin
The 48 actions of GL(2,3) on a 3x3 coordinate-array A are illustrated above. The matrices shown right-multiply the elements of A, where
Actions of GL(2,p) on a pxp coordinate-array have the same sorts of symmetries, where p is any odd prime.
It is well known that the quaternion group
is a subgroup of GL(2,3), the general linear group on the 2-space over
GF(3), the 3-element Galois field.
The figures below illustrate this fact. (Here the "2" of the note above is replaced by its equivalent, modulo 3: "-1.")
| From John Baez,
"This Week's Finds in
Mathematical Physics (Week 198),"
September 6, 2003:
Noam Elkies writes to John Baez:
From the website of
This translation plane is defined by a spreadset in a 2-dimensional vector space over the field GF(3), consisting of the following matrices.