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Generix never gives up. (English) Zbl 1103.03035
In a group of finite Morley rank, an almost self-normalizing connected nilpotent definable subgroup is called a Carter subgroup; by [O. Frécon and E. Jaligot, J. Group Theory 8, 623–633 (2005; Zbl 1083.20027)] these always exist. The author calls a definable subgroup of a stable group generous if the union of its conjugates is generic; he shows that in any group of finite Morley rank generous Carter subgroups are conjugate and generically disjoint. He conjectures that in a group of finite Morley rank Carter subgroups are always generous (and hence conjugate).
The proof is by rank computations and connectivity arguments. Note that for soluble groups of finite Morley rank, O. Frécon [J. Group Theory 9, 361–367 (2006; Zbl 1165.20023)] has shown generosity and conjugacy of all Carter subgroups.

MSC:
03C45 Classification theory, stability, and related concepts in model theory
03C60 Model-theoretic algebra
20F17 Formations of groups, Fitting classes
20F18 Nilpotent groups
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References:
[1] Generix strikes again 54 pp 847– (1989) · Zbl 0685.03032
[2] The Bulletin of Symbolic Logic 71 pp 315– (2001)
[3] Model Theory at Newton Institute (2006)
[4] Groups of finite Morley rank (1994) · Zbl 0816.20001
[5] DOI: 10.1016/j.jalgebra.2003.12.027 · Zbl 1056.20020
[6] Journal of Group Theory (2006)
[7] Journal of Group Theory 8 pp 623– (2005)
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