On \(\omega \)-categorical, generically stable groups
type: Item
instance of: scholarly article
zbMATH Open document ID: 1250.03054
author: Krzysztof Krupiński
author: Jan Dobrowolski
publication date: 2012-09-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1108.3481
full work available at URL: https://projecteuclid.org/euclid.jsl/1344862173
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 03C35
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 20A15
Mathematics Subject Classification ID: 20F19
zbMATH DE Number: 6083964
zbMATH Keywords: \(\aleph_0\)-categorical group
zbMATH Keywords: generically stable type
zbMATH Keywords: soluble subgroup of finite index
MaRDI profile type: Publication
OpenAlex ID: W2107022254
arXiv ID: 1108.3481
cites work: Totally categorical groups and rings
cites work: Superrosy dependent groups having finitely satisfiable generics
cites work: Supersimple ω-categorical groups and theories
cites work: On NIP and invariant measures
cites work: Groups, measures, and the NIP
cites work: On \(\omega\)-categorical groups and rings with NIP
cites work: ABSOLUTELY UBIQUITOUS STRUCTURES AND N0-CATEGORICAL GROUPS
cites work: Q3961653
DOI: 10.2178/JSL/1344862173
title: On \(\omega \)-categorical, generically stable groups
published in: The Journal of Symbolic Logic
review text: \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}].
reviewed by: Cédric Milliet
description: scientific article; zbMATH DE number 6083964