Some results on Quillen's Conjecture via equivalent-poset techniques

Maximal subgroups of exceptional groups and Quillen's dimension

“…In view of these results on Quillen's conjecture, in this article, we focus on showing that the 2-extensions of the exceptional finite simple groups of Lie type in odd characteristic satisfy (QD) 2 , with a small number of exceptions. This improves the conclusions of [18] on Quillen's conjecture for = 2, and allows us to conclude then that exceptional groups of Lie type in odd characteristic different from 3 cannot be components of a minimal counterexample to the conjecture (see Corollary 1.2 below).…”

“…In the last few years, there have been further developments in the Quillen conjecture [15,16,17,18]. Recently, in [18], new tools for the study of the conjecture have been provided.…”

“…In the last few years, there have been further developments in the Quillen conjecture [15,16,17,18]. Recently, in [18], new tools for the study of the conjecture have been provided. For example, it is shown that the Aschbacher-Smith general approach to the conjecture can be extended to every prime by reducing reliance on results of [1] stated only for odd primes and invoking the Classification.…”

“…For example, it is shown that the Aschbacher-Smith general approach to the conjecture can be extended to every prime by reducing reliance on results of [1] stated only for odd primes and invoking the Classification. In particular, Theorem 1.1 of [18] shows that the Main Theorem of [1] extends to ≥ 3, keeping the additional constraint on the unitary groups. On the other hand, for = 2, one important obstruction for this extension is the lack of a (QD)-List for this prime.…”

“…On the other hand, for = 2, one important obstruction for this extension is the lack of a (QD)-List for this prime. Roughly, Corollary 1.8 of [18] concludes that a minimal counterexample to Quillen's conjecture contains a component of Lie type in characteristic ≠ 3, and every component of has a 2-extension failing (QD) 2 .…”