See David G. Poole, "The Stochastic Group,"
American Mathematical Monthly, volume 102, number 9
(November, 1995), pages 798–801.
* This post was suggested by the phrase "The Diamond Theorem,
also known as the von Neumann-Birkhoff conjecture" in a
ChatGPT-3.5 hallucination today.
That phrase suggests a look at the Birkhoff-von Neumann theorem:
The B.-von N. theorem suggests a search for analogous results
over finite fields. That search yields the Poole paper above,
which is related to my own "diamond theorem" via affine groups.