Superstable groups of finite rank without pseudoplanes (Q1086228)

The author conjectures in this paper that every stable group whose theory does not type-interpret a pseudoplane (*) is abelian-by-finite and proves the conjecture for superstable groups of finite Morley rank. Since then he and \textit{U. Hrushovski} have proved the whole conjecture [Logic Colloquium '85, 233-244 (1987)]. In their paper there is the proof that a theory satisfies (*) if and only if it is weakly normal (also called: 1-based), which is a notion of pure stability theory and it is in fact this characterization of (*) which is used here. Examples of weakly normal theories are: \(\omega\)-categorical superstable theories [\textit{G. Cherlin}, \textit{L. Harrington}, \textit{A. H. Lachlan}, Ann. Pure Appl. Logic 28, 103-135 (1985; Zbl 0566.03022)] and superstable non \(\omega\)-stable unidimensional theories [\textit{S. Buechler}, ibid. 30, 83-94 (1986)]. Thus the result given here generalizes in particular the theorem of \textit{W. Baur}, \textit{G. Cherlin} and \textit{A. Macintyre} that \(\omega\)-categorical \(\omega\)-stable groups are abelian-by-finite [J. Algebra 57, 407-440 (1979; Zbl 0401.03012)].