I had a question that I was going to ask on Math Overflow, but after some research I managed to find the answer.
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 GÃ¡bor Nagy and Petr VojtechovskÃ½, explains Moufang loops, and how the classification of non-associative Moufang Loops reduces to a question about finite simple groups.