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Definably simple groups in o-minimal structures

Authors:	Y Peterzil A Pillay S Starchenko

Institution:	Department of Mathematics and Computer Science, Haifa University, Haifa, Israel ; Department of Mathemetics, University of Illinois at Urbana-Champaign, 1409 W. Green St., Urbana, Illinois 61801 ; Department of Mathemetics, University of Notre Dame, Room 370, CCMB, Notre Dame, Indiana 46556

Abstract:	Let $\mathbb{G} =\langle G, \cdot\rangle$ be a group definable in an o-minimal structure $\mathcal{M}$ . A subset of is $\mathbb{G}$ -definable if is definable in the structure $\langle G,\cdot\rangle$ (while definable means definable in the structure $\mathcal{M}$ ). Assume $\mathbb{G}$ has no $\mathbb{G}$ -definable proper subgroup of finite index. In this paper we prove that if $\mathbb{G}$ has no nontrivial abelian normal subgroup, then $\mathbb{G}$ is the direct product of $\mathbb{G}$ -definable subgroups $H_1,\ldots,H_k$ such that each is definably isomorphic to a semialgebraic linear group over a definable real closed field. As a corollary we obtain an o-minimal analogue of Cherlin's conjecture.

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