group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

, the trichotomy theorem divides the

finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...

simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...

groups of

characteristic 2 type In finite group theory, a branch of mathematics, a group is said to be of characteristic 2 type or even type or of even characteristic if it resembles a group of Lie type over a field of characteristic 2. In the classification of finite simple gr ...

and

rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...

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 In mathematical finite group theory, a group of GF(2)-type is a group with an involution centralizer whose generalized Fitting subgroup is a group of symplectic type . As the name suggests, many of the groups of Lie type In mathematics, specific ...

(classified by Timmesfeld and others), groups of "standard type" for some odd prime (classified by the

Gilman–Griess theorem In finite group theory, a mathematical discipline, the Gilman–Griess theorem, proved by , classifies the finite simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (al ...

and work by several others), and groups of uniqueness type, where Aschbacher proved that there are no simple groups.

References

* * * Theorems about finite groups {{algebra-stub