Stable group

Examples

• A group of finite Morley rank is an abstract group G such that the formula x = x has finite Morley rank for the model G. It follows from the definition that the theory of a group of finite Morley rank is ω-stable; therefore groups of finite Morley rank are stable groups. Groups of finite Morley rank behave in certain ways like finite-dimensional objects. The striking similarities between groups of finite Morley rank and finite groups are an object of active research.
• All finite groups have finite Morley rank, in fact rank 0.
• Algebraic groups over algebraically closed fields have finite Morley rank, equal to their dimension as algebraic sets.
• showed that free groups, and more generally torsion-free hyperbolic groups, are stable. Free groups on more than one generator are not superstable.

  The Cherlin–Zilber conjecture

• Any connected group of Morley rank 1 is abelian.
• Cherlin proved that a connected rank 2 group is solvable.
• Cherlin proved that a simple group of Morley rank 3 is either a bad group or isomorphic to PSL₂ for some algebraically closed field K that G interprets.
• showed that an infinite group of finite Morley rank is either an algebraic group over an algebraically closed field of characteristic 2, or has finite 2-rank.