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A generic identification theorem for groups of finite Morley rank. (English) Zbl 1047.03028

The paper is a contribution to the classification of infinite simple groups of finite Morley rank, and so to the solution of the classical conjecture of Cherlin and Zilber saying that such a group is a simple algebraic group over an algebraically closed field. The analysis of a possible minimal counterexample \(G\) to this conjecture is usually divided into two cases, according to whether \(G\) is of even or odd type (a distinction based on the structure of Sylow 2-subgroups of \(G\)), and specific ad-hoc techniques are used in each of these cases. The paper under review proposes a uniform approach, combining these \` \` sister” theories. In fact, an identification theorem for generic \(K^\star\)-groups \(G\) of finite Morley rank is shown, saying that, if \(D\) is a maximal \(p\)-torus in \(G\) of Prüfer rank \(\leq 3\), \(G\) equals \(\langle C^\circ_G(x): x \in D\), \(| x| = p \rangle\) and, for every element \(x\) of order \(p\) in \(K\), \(C^\circ_G(x)\) is of \(p'\)-type and coincides with \(F^\circ(C^\circ_G(x)) E(C^\circ_G(x))\), then \(G\) is a Chevalley group over an algebraically closed field of characteristic \(\not = p\). Notably, the proof avoids any reference to Tits’s classification of buildings, it refers instead to Lyons’s Theorem.

MSC:

03C60 Model-theoretic algebra
20G15 Linear algebraic groups over arbitrary fields
20E32 Simple groups
03C45 Classification theory, stability, and related concepts in model theory
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