On \(\omega \)-categorical, generically stable groups
\textit{W. Baur, G. Cherlin} and \textit{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 [\textit{E. Hrushovski} and \textit{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 \textit{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), \url{arXiv:1202.2327}].
scientific article; zbMATH DE number 6083964