Subgroups generated by root elements in groups of Lie type (Q1327598)

Subgroups generated by long root elements in finite groups of Lie type have been classified by the combined efforts of \textit{W. M. Kantor} for the classical groups [Trans. Am. Math. Soc. 248, 347-379 (1979; Zbl 0406.20040)], and Cooperstein for the exceptional groups (see several references in the reviewed paper). Here this work is put in a more general context, by considering the corresponding results for algebraic groups. These are relatively simple to state, as far fewer cases arise, and perhaps also relatively simple to prove, as there is more machinery available. Essentially, a (simple closed connected) subgroup generated by long root elements in an algebraic group is either a subsystem group, or a well-known large subgroup of a subsystem group. Next come some technical results relating the structure of the finite groups of Lie type to the structure of the algebraic groups. These would appear to enable the assiduous reader to deduce the main results of Kantor and Cooperstein. At the end of the paper there is also a section devoted to the corresponding results for Lie algebras, followed by a few miscellaneous results arising from the proof, which may be of independent interest.