Stable group
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).
Examples
- A group of finite Morley rank is an abstract finite-dimensionalobjects. 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 sets.
- superstable.
The Cherlin–Zilber conjecture
The Cherlin–Zilber conjecture (also called the algebraicity conjecture), due to Gregory
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
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.
References
- MR 1452677
- Altinel, Tuna; Borovik, Alexandre V.; Cherlin, Gregory (2008), Simple groups of finite Morley rank, Mathematical Surveys and Monographs, vol. 145, Providence, R.I.: MR 2400564
- Borovik, A. V. (1998), "Tame groups of odd and even type", in Carter, R. W.; Saxl, J. (eds.), Algebraic Groups and their Representations, NATO ASI Series C: Mathematical and Physical Sciences, vol. 517, Dordrecht: Kluwer Academic Publishers, pp. 341–366
- Borovik, A. V.; MR 1321141
- Burdges, Jeffrey (2007), "The Bender method in groups of finite Morley rank" (PDF), J. Algebra, 312 (1): 33–55, S2CID 9031997
- Cherlin, G. (1979), "Groups of small Morley rank", Ann. Math. Logic, 17 (1–2): 1–28,
- Pillay, Anand (2001) [1994], "Group of finite Morley rank", Encyclopedia of Mathematics, EMS Press
- Poizat, Bruno (2001), Stable groups, Mathematical Surveys and Monographs, vol. 87, Providence, RI: American Mathematical Society, pp. xiv+129, MR 1827833(Translated from the 1987 French original.)
- Scanlon, Thomas (2002), "Review of "Stable groups"", Bull. Amer. Math. Soc., 39 (4): 573–579,
- Bibcode:2006math......9096S
- Wagner, Frank Olaf (1997), Stable groups, Cambridge University Press, ISBN 0-521-59839-7
- MR 0441720