On \(\omega \)-categorical, generically stable groups. (English) Zbl 1250.03054

W. Baur, G. Cherlin and A. J. Macintyre have established that an \(\aleph_0\)-categorical stable group has a definable nilpotent subgroup of finite index (see [J. Algebra 57, 407–440 (1979; Zbl 0401.03012)]). The paper under review deals with \(\aleph_0\)-categorical groups satisfying an additional assumption, weaker than stability, introduced in [E. Hrushovski and A. Pillay, J. Eur. Math. Soc. (JEMS) 13, No. 4, 1005–1061 (2011; Zbl 1220.03016)]: generical stability. The main result establishes that an \(\aleph_0\)-categorical generically stable group has a definable solvable subgroup of finite index. To this aim, the authors prove a descending chain condition on uniformly definable subgroups with parameters in a Morley sequence of a generically stable type, and use the classification result of characteristically simple \(\aleph_0\)-categorical groups by J. S. Wilson [Lond. Math. Soc. Lect. Note Ser. 71, 345–358 (1982; Zbl 0497.20022)]. The authors conjecture that an \(\aleph_0\)-categorical generically stable group should have a definable nilpotent subgroup of finite index. The conjecture is now claimed in a recent paper by the same authors [“On \(\omega\)-categorical, generically stable groups and rings” (2012), arXiv:1202.2327].


03C45 Classification theory, stability, and related concepts in model theory
03C35 Categoricity and completeness of theories
03C60 Model-theoretic algebra
20A15 Applications of logic to group theory
20F19 Generalizations of solvable and nilpotent groups
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