New results on intermediate subgroups related to Aschbacher classes
Nikolai Vavilov (Belfast and St. Petersburg)
Wednesday 30th January, 2008 16:00-17:00 214
In 1987 Aschbacher proved a remarkable subgroup structure theorem for classical groups, which essentially asserts, that every maximal subgroup of a (finite or algebraic) classical group either belongs to one of the 8 classes C_1,...,C_8, explicitly described by their action on the natural module, or is an almost simple group in an irreducible representation. After that (and, actually, also BEFORE that) a lot of effort was invested in determining, whether (and when) subgroups from Aschbacher classes are indeed maximal. For the finite case this was completely determined (modulo the Classification of finite simple groups) by Kleidman and Liebeck. A lot of work in this direction for ARBITRARY fields was done by Dye, King, and Li Shangzhi, among others. In the 1970-ies Borewicz and his (then) students started the study of similar problems over rings. Of course, for rings you cannot expect these subgroups to be maximal anymore, but they are still fairly large, and one can reasonably expect, that a complete descrption of their overgroups might still be possible. Much of their work can be retrospectively described as the study of the classes C_1+C_2 (reducible subgroups) and, very partially, C_3 (ring extension subgroups). Little has been known for other classes. In the last 5 years we returned to the study of overgroups for other Aschbacher classes. Using a combination of geometric, ring-theoretic and K-theoretic techniques (and sometimes invoking results for fields) we succeeded in proving standard description of overgroups for most of the remaining cases, and, sometimes, in constructing very unexpected counter-examples to the standard desciption. I'm going to state the (essentially) complete answers for the cases of 1) C_8 (classical subgroups), by Petrov, myself and Hong You, 2) C_5 (subring subgroups), by Stepanov, 3) C_4+C_7 (tensored subgroups), by Ananievsky, Sinchuk, and myself, and discuss some further related results and open problems.