Finite simple groups of parabolic characteristic p

• Applicant: Professor Dr. Gernot Stroth
• Subject Area: Mathematics
• Term: from 2015 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 283297171

There are actually two different approaches to revise the classification of the finite simple groups. One due to G. Gorenstein, R. Lyons and R. Solomon (GLS) which will give a proof following the classical line. Recently M. Aschbacher started a new approach with the help of fusion systems. So far the latter approach just work for the groups of component type. Parallel there is the work of U. Meierfarnkenfeld, B. Stellmacher and G. Stroth (MSS) which aim to gve a new classification of the groups of local characteristic p. The main aim of this project is bulid a bridge between the results of MSS and the other two projects such that they can be used there.In the original lassification there was a clear subdivision in groups of component type and groups of local characteristic 2.GLS now speaks of groups of even type instead of groups of local characteristic 2. Aschbacher in his program reduces the problem to consider groups in which local subgroups containing a maximal abelian subgroup are of characteristic 2. A key could be a result due to K. Magaard and G. Stroth which classifies the groups which are of even type but not of parabolic characteristic 2. To make the results of MSS accessible for GLS it would suffice to prove these results for parabolic characteristic 2, i.e. local subgroups containing a Sylow 2-subgroup are of local characteristic 2. As a side effect this would also give the bridge to use the results in the Aschbacher program.The aim of this project is to achieve exactly this goal. More precise we will prove the structure theorem in MSS for groups of parabolic characteristic 2. Furthermore identifying the simple groups showing up should also be done by just assuming parabolic characteristic 2. In particular we have to deal with groups $G$ containing a subgroup $H$ of odd index, which is an automorphism group of a group of Lie type in characteristic two. Just assuming parabolic characteristic 2 we will determine all such pairs (G,H).

• DFG Programme: Research Grants