Some modest additions to the works of Messieurs Mark DeBonis, Franz Delahan, David Epstein and Ali Nesin on finite Morley rank Frobenius groups.
type: Item
instance of: scholarly article
title: Some modest additions to the works of Messieurs Mark DeBonis, Franz Delahan, David Epstein and Ali Nesin on finite Morley rank Frobenius groups.
author: Bruno Poizat
published in: Model Theory
publication date: 2024-07-18T00:00:00Z
review text: In the article under review, the author extends work on Frobenius groups to the case of groups of finite Morley rank. In particular, he shows (Proposition 6.4) that, given a connected group \(F\) of finite Morley rank having a definable malnormal commutative subgroup \(T\) which has same Morley rank as his Frobenius complement, the group \(F\) is definably isomorphic to a semi-direct product of the multiplicative and the additive subgroup of an algebraically closed field, interpretable in the group \(F\).\N\NRecall that a group \(F\) is a Frobenius group if it acts transitively as permutations on a finite set in such a way that the stabilizer of two points is the trivial subgroup. In particular, the stabilizer \(T\) of a point is a malnormal subgroup, which means that for any \(g\notin T\), the subgroup \(T\cap g\cdot T\) is trivial. \textit{G. Frobenius} [Berl. Ber. 1901, 1216--1230 (1901; JFM 32.0137.01)] showed that \(F\) is a semi-direct product of \(T\) and a normal subgroup \(U(T)\), called the Frobenius Kernel. Moreover, Thompson proved that \(U(T)\) is nilpotent.\N\NThe author now considers a group \(F\) of finite Morley rank having a definable malnormal subgroup \(T\), so the action of \(G\) on the set of classes modulo \(T\) is a transitive action such that the stabilizer of two classes is the trivial subgroup. Whilst the index of \(T\) need not be finite, the author uses some results due to \textit{A. Borovik} and \textit{A. Nesin} [Groups of finite Morley rank. Oxford: Clarendon Press (1994; Zbl 0816.20001)] to show (Proposition 4.2) that a pseudo-locally finite Frobenius group of finite Morley rank is a semi-direct product of \(T\) and \(U(T)\). Moreover, the kernel \(U(T)\) is commutative whenever \(T\) contains an involution. In Propositions 5.1 and 5.4, the author gives some rank estimates for a Frobenius group of finite Morley rank and a description of its socle.
reviewed by: Amador Martin-Pizarro
Mathematics Subject Classification ID: 20F11
Mathematics Subject Classification ID: 14L99
Mathematics Subject Classification ID: 20E32
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 03C60
zbMATH DE Number: 7882661
zbMATH Keywords: model theory
zbMATH Keywords: Frobenius groups
zbMATH Keywords: groups of finite Morley rank
MaRDI profile type: Publication
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DOI: 10.2140/MT.2024.3.71
description: scientific article; zbMATH DE number 7882661