In mathematical finite group theory, the trichotomy theorem divides the simple

Simple group

In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...

 groups of characteristic 2 type and rank at least 3 into three classes. It was proved by for rank 3 and by for rank at least 4.
The three classes are groups of GF(2) type (classified by Timmesfeld and others), groups of "standard type" for some odd prime (classified by the Gilman–Griess theorem

Gilman–Griess theorem

In finite group theory, a mathematical discipline, the Gilman–Griess theorem, proved by , classifies the finite simple groups of characteristic 2 type with e ≥ 4 that have a "standard component", which covers one of the three cases of the trichotomy theorem....

 and work by several others), and groups of uniqueness type

Uniqueness case

In mathematical finite group theory, the uniqueness case is one of the three possibilities for groups of characteristic 2 type given by the trichotomy theorem....

, where Aschbacher proved that there are no simple groups.