Eliminating components in Quillen's Conjecture

TL;DR

This paper proves that in minimal counterexamples to Quillen's conjecture, every component must admit an outer automorphism, leading to the elimination of certain components and narrowing the possible structures involved.

Contribution

It generalizes Segev's result by showing all components in a minimal counterexample admit outer automorphisms, enabling stronger eliminations of certain group components.

Findings

Eliminates sporadic and alternating components for odd p

Reduces the problem to Lie-type components with p-outer automorphisms

Provides tools for eliminating components with p-outer automorphisms

Abstract

We generalize an earlier result of Segev, which shows that {\em some\/} component in a minimal counterexample to Quillen's conjecture must admit an outer automorphism. We show in fact that {\em every\/} component must admit an outer automorphism. Thus we transform his restriction-result on components to an elimination-result: namely one which excludes any component which does not admit an outer automorphism. Indeed we show that the outer automorphisms admitted must include $p$-outers: that is, outer automorphisms of order divisible by $p$. This gives stronger, concrete eliminations: for example if $p$ is odd, it eliminates sporadic and alternating components -- thus reducing to Lie-type components (and typically forcing $p$-outers of field type). For $p = 2$, we obtain similar but less restrictive results. We also provide some tools to help eliminate suitable components that do admit $p$-outers in a minimal counterexample.