Conjecture A simple group of finite Morley rank is algebraic.
On this page we explain the Borovik Program
The four types | K*-groups | Mixed and even type | Odd and degenerate type |
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 |
---|---|---|---|
S^{o}=U Unipotent |
S^{o}=T Torus |
S^{o}=U*T | S^{o}=1 S Finite |
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.
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.
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.)
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.
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.)
Reference: Borovik/Nesin, Groups of Finite Morley Rank, Oxford Logic Guides 26.
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Alexandre Borovik Tuna Altinel Eric Jaligot |