Stable group

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In model theory, a stable group is a group that is stable in the sense of stability theory. An important class of examples is provided by groups of finite Morley rank (see below).


The Cherlin–Zilber conjecture[edit]

The Cherlin–Zilber conjecture (also called the algebraicity conjecture), due to Gregory Cherlin (1979) and Boris Zil'ber (1977), suggests that infinite (ω-stable) simple groups are simple algebraic groups over algebraically closed fields. The conjecture would have followed from Zilber's trichotomy conjecture. Cherlin posed the question for all ω-stable simple groups, but remarked that even the case of groups of finite Morley rank seemed hard.

Progress towards this conjecture has followed Borovik’s program of transferring methods used in classification of finite simple groups. One possible source of counterexamples is bad groups: nonsoluble connected groups of finite Morley rank all of whose proper connected definable subgroups are nilpotent. (A group is called connected if it has no definable subgroups of finite index other than itself.)

A number of special cases of this conjecture have been proved; for example:

  • 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(K) for some algebraically closed field K that G interprets.
  • Tuna Altinel, Alexandre V. Borovik, and Gregory Cherlin (2008) 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.