Abnormal subgroups in solvable groups of finite Morley rank
The author begins a systematic study of the theory of formations in connected soluble groups of finite Morley rank. Recall that a subgroup \(H\) of a group \(G\) is abnormal if \(g\in\langle H,H^g\rangle\) for any \(g\in G\); it is a Carter subgroup if it is nilpotent and self-normalizing. The author proves that in a connected soluble group of finite Morley rank minimal abnormal subgroups exist and are conjugate; they are precisely the Carter subgroups (Théorème 1.1). Moreover, the following are equivalent: (1) \(H\) is abnormal. (2) \(H\) contains a Carter subgroup. (3) \(H\) is definable, and any \(g\in G\) is contained in the smallest definable supergroup of \(H\cup H^g\). (4) \(H\) is definable, connected, and there is a sequence \(G=H_0>H_1>\cdots>H_n=H\) such that \(H_{i+1}\) is a proper connected definable non-normal subgroup of \(H_i\) for all \(i<n\). (Théorème 1.2.) This extends, in the context of groups of finite Morley rank, earlier work of \textit{F. Wagner} [Arch. Math. Logic 33, No. 1, 23-34 (1994; Zbl 0813.03022)], who proved existence and conjugacy for Carter subgroups in the larger class of ``very'' connected small stable groups. In a final section, the author develops a theory of connected saturated formations for soluble groups of finite Morley rank; for such a formation, covering subgroups exist and are conjugate (Théorème 8.5). By his previous results, the class of connected nilpotent groups of finite Morley rank is a connected saturated formation, whose covering subgroups are precisely the Carter subgroups. It is to be expected that these results will have many applications in the study of soluble groups of finite Morley rank. A first application is by \textit{E. Jaligot} [J. Algebra 240, No. 2, 413-444 (2001; Zbl 0994.20028)]. In the introduction the author claims that B. Poizat has constructed a bad field. This is not quite true: \textit{B. Poizat} [J. Symb. Log. 64, No. 3, 1339-1355 (1999; Zbl 0938.03058)], and \textit{J. T. Baldwin} and \textit{K. Holland} [J. Symb. Log. 65, No. 1, 371-391 (2000; Zbl 0957.03044)] have constructed an impure field of Morley rank \(2\), but not a field of finite Morley rank with a proper infinite multiplicative subgroup.
scientific article; zbMATH DE number 1496007
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