Some modest additions to the works of Messieurs Mark DeBonis, Franz Delahan, David Epstein and Ali Nesin on finite Morley rank Frobenius groups.
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.
scientific article; zbMATH DE number 7882661
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