Stable group

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In model theory, a stable group is a group that is stable in the sense of stability theory. In particular any group of finite Morley rank is stable, where a group of finite Morley rank is an abstract group that behaves in certain ways like a finite-dimensional object. More precisely, it is a 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.

[edit] Examples

• 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.
• Sela (2006) showed that free groups, and more generally torsion free hyperbolic groups, are stable. Free groups on more than one generator are not superstable.

[edit] The Cherlin-Zilber conjecture

The Cherlin-Zilber conjecture, due to (Cherlin 1979), and Boris Zilber (1977), suggests that (ω-stable) simple groups of finite Morley rank are simple algebraic groups over algebraically closed fields. ((Cherlin 1979) asked a similar question for all ω-stable simple groups, and remarked that even the case of groups of finite Morley rank seemed hard.)

Much of the work on this problem is Borovik’s program of transferring methods from finite group theory. This method depends on the 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 not 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 algebraicially closed field K that G interprets.

[edit] References

• Altinel, Tuna; Borovik, Alexandre & Cherlin, Gregory (1997), “Groups of mixed type”, J. Algebra 192 (2): 524-571, MR1452677, DOI 10.1006/jabr.1996.6950 
• Borovik, A. V. (1998), “Tame groups of odd and even type”, Algebraic Groups and their Representations, vol. 517, NATO ASI Series C: Mathematical and Physical Sciences, Dordrecht: Kluwer Academic Publishers, pp. 341–366 
• Borovik, A. V. & Nesin, A. (1994), Groups of Finite Morley Rank, vol. 26, Oxford Logic Guides, New York: Oxford University Press, MR1321141, ISBN 0-19-853445-0 
• Burdges, Jeffrey (2007), “The Bender method in groups of finite Morley rank”, J. Algebra 312 (1): 33–55, MR2320445, <http://www.math.rutgers.edu/~burdges/BenderMethod.pdf> 
• Cherlin, G. (1979), “Groups of small Morley rank”, Ann. Math. Logic 17: 1–28, DOI 10.1016/0003-4843(79)90019-6 
• Pillay, A. (2001), “Group of finite Morley rank”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• Poizat, Bruno (2001), Stable groups, vol. 87, Mathematical Surveys and Monographs, Providence, RI: American Mathematical Society, pp. xiv+129, MR1827833, ISBN 0-8218-2685-9  (Translated from the 1987 French original.)
• Scanlon, Thomas (2002), “Review of "Stable groups"”, Bull. Amer. Math. Soc. 39: 573-579, <http://www.ams.org/bull/2002-39-04/S0273-0979-02-00953-9/> 
• Wagner, Frank Olaf (1997), Stable groups, Cambridge University Press, ISBN 0521598397 
• Zil'ber, B. I. (1977), “Группы и кольца, теория которых категорична (Groups and rings whose theory is categorical)”, Fundam. Math. 95: 173-188, MR0441720, <http://matwbn.icm.edu.pl/tresc.php?wyd=1&tom=95&jez=>