# Algebraicity conjecture

## Cherlin/Zilber

Conjecture   A simple group of finite Morley rank is algebraic.

 The four types K*-groups Mixed and even type Odd and degenerate type

## 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 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

Or more explicitly:
Even Odd Mixed Degenerate
So=U
Unipotent
So=T
Torus
So=U*T So=1
S Finite

## 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.

A minimal counterexample to the Algebraicity Conjecture would be a K*-group; conversely, a K*-group is either a K-group or a minimal counterexample to the algebraicity conjecture.

## 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).

### Mixed and Even Type

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

More generally, there are no simple groups of finite Morley rank of mixed type, and any simple group of finite Morley rank is algebraic.

(Details, mixed and even type.)

### Odd Type

In the odd type case, any simple K*-group of finite Morley rank is either algebraic, or is minimal connected simple with Prüfer 2-rank at most two.

### Degenerate Type

In the degenerate K* case, the normal 2-rank is at most 2 or there is a proper 2-generated core.

(Details, odd and degenerate type.)

## Surveys

Overview

• A Brief Introduction to groups of Finite Morley Rank
• A. Borovik; based on a talk given in 1994
21 pp. Introductory. Tame conjecture in Section 4
• Tame groups of Odd and Even Type
• A. Borovik, November 1997; appeared 1998 in Carter/Saxl Algebraic Groups and their Representations, Kluwer
27 pp. Full statement of the program. Even type case was expected to use component analysis. Most of this was bypassed.
• Simple groups of finite Morley rank, 2002
• G. Cherlin, Spring 2002, Ravello. To appear 2005 in proceedings volume.
17 pp. Emphasizes Burdges' work eliminating tameness hypotheses in odd type (somewhat oversimplified, cf. Burdges' dissertation, Rutgers 2004, Chap. 10) and Altinel's approach to even and mixed type, then at an early stage.

Even Type

• Simple groups of finite Morley rank of even type
• Tuna Altinel, Fall 2000 (this copy shows a printing date of 2002)
22 pp. The classification of even type K*-groups.
• Classification of the simple groups of finite Morley rank
• Tuna Altinel, Contemporary Mathematics 302, 2002, 121-147
26 pages. After discussing even type, also enters into the theory of solvable groups, degenerate type (and relations to Feit-Thompson) as well as his (new) approach to even type without K* hypotheses.

## Survey Talks (Slides only)

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