On groups of hyperbolic length
This paper concerns the function \(\ell(G)\), the length of the longest chain of subgroups of \(G\), for \(G\) a finite Lie type group. Such a group \(G\) is said to be of hyperbolic length if no chain of maximal length for \(G\) can include a parabolic subgroup. Using detailed information on the maximal subgroups of \(G\), the authors prove that to calculate \(\ell(G)\), one need consider only some very special types of maximal subgroups. In particular, they obtain that the maximal parabolic subgroups are almost always enough. Their results imply that, for each prime \(p\), there exist only finitely many (possibly \(0\)) finite Lie type groups \(G\) in characteristic \(p\) with \(G\) of hyperbolic length. This strengthens a result of \textit{R. Solomon} and \textit{A. Turull} [J. Lond. Math. Soc., II. Ser. 44, No. 3, 437-444 (1991; Zbl 0776.20007)]. (Also submitted to MR).
scientific article; zbMATH DE number 1033780
scientific article