A group of finite Morley rank is a group $ (G,\cdot) $, usually with extra structure, whose Morley rank is less than $ \omega $.
The Cherlin-Zilber conjecture asserts that every simple group of finite Morley rank is an algebraic group over a field. This remains open as of 2014.
However, a considerable amount is known about groups of finite Morley rank. See for example, Bruno Poizat's book Stable Groups, as well as…[more recent books]
- Morley rank and Lascar rank coincide, and are definable. In particular, Morley rank satisfies the Lascar inequalities.
- If $ G $ is a group of finite Morley rank, then the connected component $ G^0 $ exists, and is definable, rather than merely being type-definable. There is a unique type in $ G^0 $ of maximal Morley rank, i.e., $ G^0 $ has Morley degree 1. The translates of $ G^0 $ are called the generics of $ G $, and have many good properties. They are the unique types which are translation invariant.
- Any field of finite Morley rank is algebraically closed, but may have additional structure.
- A group of finite Morley rank is simple (in the group theoretic sense) if and only if it is definable simple. That is, if $ G $ is not simple as an abstract group, then $ G $ has a definable normal subgroup.
- Every infinite group of finite Morley rank contains an infinite abelian definable subgroup.
- Every simple group of finite Morley rank is almost strongly minimal, i.e., is algebraic over a strongly minimal set.
- Groups of finite Morley rank are "dimensional." This falls out of the Lascar analysis.
- Every type-definable subgroup of a group of finite Morley rank is, in fact, definable.
Transitive action on a strongly minimal set Edit
One rather strong result about groups of finite Morley rank is the following:
Let $ G $ be a group of finite Morley rank, acting transitively and faithfully on a strongly minimal set $ S $. Then we are in one of the following three situations:
- $ G $ has rank 1, is commutative, and $ S $ is a $ G $-torsor.
- $ G $ has rank 2, $ S $ is the affine line over a definable field $ K $, and $ G $ is the group of affine linear transformations over $ K $
- $ G $ has rank 3, $ G $ is $ PSL_2(K) $ for a definable field $ K $, and $ S $ is the projective line over $ K $, with the usual action.
In cases 2 or 3, $ K $ is algebraically closed. $ G $ cannot have rank greater than 3.
Under the hypothesis that there are no bad groups, it can be shown that this implies that the Cherlin-Zilber conjecture holds for groups of Morley rank at most 3: any simple group of Morley rank at most 3 must be $ PSL_2(K) $ for a definable field $ K $.
It also implies that if $ G $ is a simple group of finite Morley rank, containing a definable subgroup $ H $ such that $ RM(H) = RM(G) - 1 $, then $ G $ has rank 3 and is $ PSL_2(K) $ over an algebraically closed definable field.