Minimal bounded index subgroup for dependent theories. (English) Zbl 1144.03026
A theory is dependent if no formula \(\phi(x,y)\) has the independence property (which requires the existence of elements \(\{a_i:i\in\omega\}\) and \(\{b_I:I\subseteq\omega\}\) such that \(\models\phi(a_i,b_I)\) iff \(i\in I\)). The author proves that in a dependent theory every type-definable group has a minimal type-definable subgroup of bounded index (which must be unique and normal).
This had been proven before by E. Hrushovski, Y. Peterzil and A. Pillay under the assumption of the existence of an invariant measure. The current version also appears in their paper [J. Am. Math. Soc. 21, No. 2, 563–596 (2008; Zbl 1134.03024)]. The paper is written in typical Shelah style; I strongly recommend the version by Hrushovski, Peterzil and Pillay.
This had been proven before by E. Hrushovski, Y. Peterzil and A. Pillay under the assumption of the existence of an invariant measure. The current version also appears in their paper [J. Am. Math. Soc. 21, No. 2, 563–596 (2008; Zbl 1134.03024)]. The paper is written in typical Shelah style; I strongly recommend the version by Hrushovski, Peterzil and Pillay.
Reviewer: Frank Wagner (Villeurbanne)
MSC:
| 03C45 | Classification theory, stability, and related concepts in model theory |
| 03C60 | Model-theoretic algebra |
Citations:
Zbl 1134.03024References:
| [1] | J. T. Baldwin and Jan Saxl, Logical stability in group theory, J. Austral. Math. Soc. Ser. A 21 (1976), no. 3, 267 – 276. · Zbl 0342.02036 |
| [2] | Alessandro Berarducci, Margarita Otero, Yaa’cov Peterzil, and Anand Pillay, A descending chain condition for groups definable in o-minimal structures, Ann. Pure Appl. Logic 134 (2005), no. 2-3, 303 – 313. · Zbl 1068.03033 · doi:10.1016/j.apal.2005.01.002 |
| [3] | Ehud Hrushovski, Ya’acov Peterzil, and Anand Pillay. Groups, measures, and the NIP. preprint, 2005. · Zbl 1134.03024 |
| [4] | Saharon Shelah. Definable Groups for Dependent Theories. · Zbl 1195.03040 |
| [5] | Saharon Shelah. Dependent first order theories, continued. Israel Journal of Mathematic, accepted. math.LO/0406440. · Zbl 1195.03040 |
| [6] | Saharon Shelah. Strongly dependent theories. Israel Journal of Mathematics, submitted. math.LO/0504197. · Zbl 1195.03040 |
| [7] | Shelah, Saharon. Representation over orders of elementary classes. · Zbl 0639.03034 |
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