A group of finite Morley rank is a group (G, ⋅ ), usually with extra structure, whose Morley rank is less than ω.
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 G0 exists, and is definable, rather than merely being type-definable. There is a unique type in G0 of maximal Morley rank, i.e., G0 has Morley degree 1. The translates of G0 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
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 PSL2(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 PSL2(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 PSL2(K) over an algebraically closed definable field. ::: ::: :::