Timeline for Definition of "finite group of Lie type"?
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| Jul 17, 2013 at 13:30 | comment | added | Jim Humphreys | @Derek: I'd form $\mathcal{L}_1$ by starting with $G^F$ for (generalized) Frobenius morphisms, then do usual group-theoretic things (to get the simple groups including Suzuki/Ree, in particular). But I don't see how to limit reductie $G$ to allow general linear groups but not direct products with a multiplicative group. (In representation theory one needs tori over finite fields: are these of Lie type?) I was motivated partly by a recent paper of Brunat-Lubeck: front.math.ucdavis.edu/1211.3692. Given the simple groups in (2), I also don't see how to recover (1) by group theory alone. | |
| Jul 17, 2013 at 2:24 | comment | added | user36938 | How about defining $\mathcal{L}_1$ to be the set of non-commutative quotients of $G(k)$ for connected semisimple group $G$ over finite fields $k$ such that $G$ is absolutely simple over $k$? This includes spin groups (not Pin groups) -- they should be regarded as classical -- and away from a few types with $k$ of size 2 or 3 the derived group of $G(k)$ is a central quotient of the group $\widetilde{G}(k)$ that is perfect due to BN-pair arguments ($\widetilde{G}$ is the simply connected central cover of $G$). It doesn't give Ree and Suzuki groups, but such is life. | |
| Jul 16, 2013 at 21:05 | history | answered | Derek Holt | CC BY-SA 3.0 |