Check out Ramsey Theory -- basically a significant generalization of the pigeonhole principle which underlies a lot of automata and formal language theory (or should I say, the pigeonhole principle is the simplest case of Ramsey Theory). It basically says that even highly disordered structures turn out to necessarily contain a lot of order if they are sufficiently large. For a small example just beyond the pigeonhole principle, note that if you take any six people, then either three of them mutually know each other or three of them mutually do not know each other.
This paper looks like a nice place to start for connections with computer science, but you can google for more. It's more combinatoric than algebraic in its basic nature, but has many applications in algebra and theoretical CS.
And also check out the story of the inventor, Frank Ramsey -- truly a remarkable polymath who made fundamental, even revolutionary contributions in economics and philosophy as well as mathematics, many unappreciated until much later, all before dying at the age of 26 -- just think! In fact, Ramsey's original theorem, the basis of Ramsey Theory, was a mere lemma in a paper with a bigger aim in mathematical logic.