2
$\begingroup$

Tits has proved that a finite simple group $G$ with a BN-pair of rank $n \ge 3$, is of Lie type. Let $B$ be the Borel subgroup and $(W,S)$ the Coxeter system. The subset lattice of the set $S$ is lattice-equivalent to the interval $[B,G]$ of parabolic subgroups, so is equivalent to the boolean lattice of rank $n$.

We ask about the following extension of Tits' theorem:
Is a finite simple group $G$ having a subgroup $H$ with $[H,G]$ boolean of rank $n \ge 3$, of Lie type?

For $|G| < 4 \cdot 10^6$ (using GAP) the only examples are ${\rm A}_3(2)$, $^2{\rm A}_2(5^2)$, ${\rm C}_3(2)$ and $^2{\rm A}_3(3^2)$.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.