On this page we explain the four types of groups of finite Morley rank, and the notion of a K*-group.

The Four Types

The connected component of a Sylow 2-subgroup of a group of finite Morley rank is a central product U*T with U 2-unipotent and T a 2-torus; the intersection of U and T is finite.

In algebraic groups the 2-elements are unipotent if the characteristic is 2, and semisimple otherwise; so in characteristic 2 one expects T=1, and otherwise one expects U=1. However, a product of finitely many algebraic groups of varying characteristics also has finite Morley rank.

Accordingly, we consider a four types of groups of finite Morley rank, corresponding to the possible structures of the Sylow 2-subgroup:

	T>1	T=1
U>1	Mixed Type	Even type
U=1	Odd Type	Degenerate Type

K-groups and K*-groups

A K-group is one whose definable infinite simple sections are algebraic.

A K*-group is one whose proper definable infinite simple sections are algebraic.

Results

The structure of the Sylow 2-subgroup in a simple K*-group of finite Morley rank has been substantially clarified. The critical distinction is between the cases U>1 (mixed and even type) and U=1 (odd and degenerate types).

In the mixed and even type cases, any simple K*-group of finite Morley rank is algebraic.

(Details, mixed and even type.)

In the odd type case, the situation is less clear. There are three cases:

• G is algebraic;
• G has Prufer 2-rank at most 2;
• G is a minimal simple group (in the sense that G has no proper definable connected simple sections);

If G is tame, this can be simplified; G is either algebraic or has Prufer 2-rank at most 2.

In the degenerate case, the normal 2-rank is at most 2.

(Details, odd and degenerate type.)

Survey

• Simple groups of finite Morley rank, 2002
G. Cherlin

Reference: Borovik/Nesin, Groups of Finite Morley Rank, Oxford Logic Guides 26.