Groups of hyperbolic length in odd characteristic groups of Lie type (Q1322525)

This paper is one of a series investigating the maximal length of a chain of subgroups in a group of Lie type defined over a field of odd order. The corresponding problem for fields of characteristic 2 is much easier, and \textit{R. Solomon} and \textit{A. Turull} have shown that in every case there is a chain of maximal length passing through a parabolic subgroup [J. Lond. Math. Soc., II. Ser. 44, 437-444 (1991; Zbl 0776.20007)]. In odd characteristic this is no longer true in general, and the author defines a group of Lie type to have hyperbolic length if no chain of maximal length passes through a parabolic subgroup. The purpose of the present paper is to get some hold on the structure of a maximal subgroup \(M\) through which a chain of maximal length passes in a group \(G\) of hyperbolic length. The main result is a reduction theorem, which says that \(M\) is either the normalizer of a central product of smaller groups of hyperbolic length, or is one of a few particular groups. Note: Reference [4] has appeared in J. Algebra 163, 739-756 (1994; Zbl 0802.20020), and [6] in J. Algebra 170, 440-469 (1994; see the following review Zbl 0817.20025).