Einfache Gruppen von parabolischer Charakteristik p.
Zusammenfassung der Projektergebnisse
Some years ago U. Meierfrankenfeld, B, Stellmacher and G. Stroth started a project to understand finite simple groups (MSS for short). One of the results was the so called Local Structure Theorem. There Kp-groups G are studied, which have the property to contain a large subgroup Q. Denote by LG (S) the set of all 2-local subgroups L of G containing a Sylow p-subgroup S of G with Q ≤ S. For such an L denote by YL the largest normal p-subgroup of L such that Op (L/CL (YL )) = 1. The Local Structure Theorem describes the action of L on YL for all L ∈ LG (S) with L ≤ NG (Q). Recently M. Aschbacher started a project to classify the finite simple groups using the theory of fusion systems. In his approach he is working with groups G, which possess a 2-local subgroups L with CG (O2 (L)) ≤ O2 (L), while in the MSS-program CG (Op (L)) ≤ Op (L) if S ≤ L. The idea is to combine both programs to end up with a new classification of the finite simple groups. The Local Structure Theorem has one part (10)(a), where the description of L as explained above was not given. Hence Chr. Parker and G. Stroth showed that for p = 2 this case does not show up. This makes the Local Structure Theorem applicable for Aschbacher’s approach. Jointly with U. Meierfrankenfeld and Chr. Parker we started a project to extend the information given by the Local Structure Theorem to get information about the groups G. We started with the situation that for some L ∈ LG (S) we have that YL ≤ Q. Then we determined NG (Q) and at least for p = 2 we determined also the group G. The third result in this project was obtained jointly with G. Pientka, Chr. Parker and A. Seidel. Here we dealt with the general problem to identfy groups of Lie type in characteristic p. The assumption is that G is a Kp -group of local characteristic p, i.e. CG (Op (L)) ≤ Op (L) for all p-local subgroups L, H is a subgroup of G, which is of Lie type in characteristic p, Lie rank at least two, CG (H) = 1 and |G : NG (H)| is coprime to p. We showed that G = NG (H) if the Lie rank is at least three. In the small cases of Lie rank two we classified all examples G = NG (H) if p = 2 and did it in most cases if p is odd. If in addition G is a K2 -group then we can determine all pairs (G, H) besides H is an automorphism group of P SL3 (3) or P SL3 (7).
Projektbezogene Publikationen (Auswahl)
- “The local structure theorem, the non-characteristic 2 case”, Proc. Lond. Math. Soc. (3) 120 (2020),465513
Chr. Parker, G.Stroth
(Siehe online unter https://doi.org/10.1112/plms.12291) - “The local structure theorem: the wreath product case”, J. Algebra 561, (2020), 374-401
Chr. Parker, G. Stroth
(Siehe online unter https://doi.org/10.1016/j.jalgebra.2019.08.013)
