Weyl groups of small groups of finite Morley rank. (English) Zbl 1207.20025
The article under review is another piece of transfer of the arguments of G. Cherlin and E. Jaligot [J. Algebra 276, No. 1, 13-79 (2004; Zbl 1056.20020)] for tame minimal connected simple groups of finite Morley rank to the general case of minimal connected simple groups (without the tameness assumption). Here it is mostly the technics for the study of the Weyl group which are used in the general case.
The main theorem of the paper states that if \(T\) is a decent torus of a minimal connected simple group, then the Weyl group \(W=N(T)/C(T)\) is cyclic and has an isomorphic lifting to \(G\), and moreover no primes dividing \(|W|\) appear in \(T\) (except possibly \(2\)). Given the classification of minimal connected simple groups of odd type as provided in full generality by A. Deloro, [J. Algebra 319, No. 4, 1636-1684 (2008; Zbl 1169.20016)], the paper mostly deals with the specific case of groups without involutions.
The main theorem of the paper states that if \(T\) is a decent torus of a minimal connected simple group, then the Weyl group \(W=N(T)/C(T)\) is cyclic and has an isomorphic lifting to \(G\), and moreover no primes dividing \(|W|\) appear in \(T\) (except possibly \(2\)). Given the classification of minimal connected simple groups of odd type as provided in full generality by A. Deloro, [J. Algebra 319, No. 4, 1636-1684 (2008; Zbl 1169.20016)], the paper mostly deals with the specific case of groups without involutions.
Reviewer: Eric Jaligot (Grenoble)
Keywords:
groups of finite Morley rank; minimal connected simple groups; Weyl groups; Cartan subgroupsReferences:
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