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Quillen's conjecture and unitary groups

Antonio Díaz Ramos

Abstract

We prove that the Quillen posets $\mathcal{A}_p(H)$ of $p$-extensions $H$ of simple unitary groups have non-zero homology in the largest possible dimension, with just a few exceptions. This establishes a conjecture raised by Aschbacher-Smith in 1992. In particular, by their work and a more recent article by Piterman-Smith, Quillen's conjecture on the p-subgroup posets holds for odd primes.

Quillen's conjecture and unitary groups

Abstract

We prove that the Quillen posets of -extensions of simple unitary groups have non-zero homology in the largest possible dimension, with just a few exceptions. This establishes a conjecture raised by Aschbacher-Smith in 1992. In particular, by their work and a more recent article by Piterman-Smith, Quillen's conjecture on the p-subgroup posets holds for odd primes.
Paper Structure (10 sections, 17 theorems, 125 equations, 4 figures)

This paper contains 10 sections, 17 theorems, 125 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Group theoretical preliminaries
  3. General and special unitary groups.
  4. Automorphisms of unitary groups.
  5. Quasi-reflections in $\operatorname{GU}\nolimits_n(q)$.
  6. Symmetric group and certain subsets of it
  7. Topological preliminaries.
  8. Some homology classes.
  9. Proof of Theorem A.
  10. Proof of Theorem B.

Key Result

Theorem 1

Let $G$ be a finite group. Assume that Then $G$ satisfies Quillen's conjecture for $p$.

Figures (4)

  • Figure 1: Triangulations of the sphere $S^2$ constructed for $\operatorname{PSL}\nolimits_4(q)$ in DiazMazza2020 (left), and for $\operatorname{PGU}\nolimits_4(q)$ in Theorem \ref{['thm:mainPGU']} (right).
  • Figure 2: Barycentric subdivision of a $2$-simplex.
  • Figure 3: Triangulation of the sphere $S^1$ arising from $S_7$.
  • Figure 4: Complex for $\operatorname{PGU}\nolimits_3(q)$ (left) and $\operatorname{PGU}\nolimits_3(q)\langle \Phi\rangle$ (right).

Theorems & Definitions (46)

  • Theorem : AS1993
  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • ...and 36 more