Groups of finite Morley rank

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
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.