Finite simple groups have a complete classification. I was wondering if there were any weakenings of the axioms of group that also allowed a complete classification of the simple objects. (Here, I mean no nontrivial quotients.) Surprisingly, there’s a classification for semigroups. In the theory of semigropus the term “Simple& is used for a weaker notion. Semigroups with no nontrivial quotients are known as “congruence-free”. The classification of finite congruence-free semigroups splits into two cases: for semigroups with a zero (an element 0 such that 0x = 0) there’s an explicit construction, while a congruence-free semigroup without a zero must be a simple group.

Another direction to generalize is weaken the form of associativity. The most-studied weakening is the Moufang property, which includes the octonions as a non-trivial example. Here, the complete classification is also known: a finite simple Moufang loop is either a group or a Paige loop, which is a non-associative construction closely related to the octonions, but defined over a finite field. It’s interesting that in this case, the one non-associative family resembles simple groups of Lie type, in that it’s parameterized by the finite fields. This classification relies non-trivially on the classification of simple groups, in that the explicit classification is used to rule out any other non-associative examples.

The paper Octonions, simple Moufang loops and triality by Gábor Nagy and Petr Vojtechovský, explains Moufang loops, and how the classification of non-associative Moufang Loops reduces to a question about finite simple groups.

Ahh, but the strawberries that’s… that’s where I had them. They laughed at me and made jokes but I proved beyond the shadow of a doubt and with… geometric logic… that a duplicate key to the wardroom icebox DID exist, and I’d have produced that key if they hadn’t of pulled the Caine out of action.

The upside of the technique is that it allows you to deduce the existence of real-world objects such as keys. The downside is that it drives you insane.

(More on the context of the quote here.)

Theo Raedschelders has written a nice sketch of Herstein’s commutativity theorem for rings. It is a generalization of Wedderburn’s theorem that a finite division ring must be a field. The theorem states that if for every pair of elements $a$ and $b$ there exists an $n > 1$ (which can depend on $a$ and $b$) such that
$$
(ab – ba)^n = ab – ba,
$$
then the ring is commutative. What’s surprising about the proof is its indirectness. The proof requires essentially all of Nathan Jacobson’s structure theory for rings.

PlanetMath has a nice summary of known conditions that imply commutativity.

Sam Nelson has a nice introduction to the general area of universal-algebraic invariants from the Notices, called Revolution in Knot Theory. Nelson also discusses the emergence of virtual knots.

Here’s a sample of tex-encoded math formulas that I copied from the MathJax site.

When $a \ne 0$, there are two solutions to \(ax^2 + bx + c = 0\) and they are $$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$

If you actually view the source for the file in your browser, you’ll see the raw tex commands in the post. MathJax uses CSS to render it on the fly. I’m actually surprised this is possible, but apparently it is.

The book is elementary, and (as translated by Townsend) is a pleasant read. Much of the book centers around constructing the field of real numbers in terms of the axiomized geometrical constructions. This in turn allows Hilbert to show that the set of axioms is complete. The topic leads naturally to one of the main themes of research in plane geometry in the early part of the last century, which is to consider different algebraic objects and how they can serve as coordinates for different notions of affine or projective planes. The reals can be replaced with an arbitrary division ring, for example. For a projective plane, the most general object is a planar ternary ring, with has a ternary operation that serves as a hybrid of addition and multiplication. Determining the projective planes with a finite number of points is still an open question.

1. How much it sounded like the kind of bizarre model you’d see on Charline on Numb3rs come up with in order to crack the case.
2. That Cosma Shalizi would hate the model, since it’s the kind of a casual use of power laws he regularly criticizes. And here’s his analysis of the paper. He points out that, as in many other cases, a lognormal distribution provides a better fit.