The Bender method in groups of finite Morley rank. (English) Zbl 1130.20030
The Bender method in the classification of finite simple groups is the analysis of maximal subgroups containing the centralizer of an involution. In this technical paper the author studies maximal non-Abelian intersections of Borel subgroups of a group of finite Morley rank. Similar to the Bender uniqueness theorem, this yields information about normalizers of various subgroups of the intersection of two distinct maximal subgroups, but there is no hypothesis even on the existence of involutions.
The proofs use in particular the author’s notion of \(0,r\)-unipotency in characteristic \(0\); the results are applied in a joint paper with G. Cherlin and E. Jaligot [J. Algebra 314, No. 2, 581-612 (2007; see the following review Zbl 1130.20031)] to show that the Prüfer rank of a simple non-algebraic \(K^*\)-group of finite Morley rank of odd type is at most two.
The proofs use in particular the author’s notion of \(0,r\)-unipotency in characteristic \(0\); the results are applied in a joint paper with G. Cherlin and E. Jaligot [J. Algebra 314, No. 2, 581-612 (2007; see the following review Zbl 1130.20031)] to show that the Prüfer rank of a simple non-algebraic \(K^*\)-group of finite Morley rank of odd type is at most two.
Reviewer: Frank Wagner (Villeurbanne)
MSC:
| 20E32 | Simple groups |
| 03C45 | Classification theory, stability, and related concepts in model theory |
| 03C60 | Model-theoretic algebra |
| 20E28 | Maximal subgroups |
| 20E07 | Subgroup theorems; subgroup growth |
Keywords:
infinite simple groups; groups of finite Morley rank; minimal simple groups; Bender method; maximal intersections; Borel subgroups; maximal subgroups; centralizersCitations:
Zbl 1130.20031References:
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