Acyclic 2-dimensional complexes and Quillen's conjecture (Q2232001)

A well-known conjecture by Quillen states that if the order complex of the poset of nontrivial \(p\)-subgroups of a finite group \(G\) is contractible, then its largest normal \(p\)-subgroup \(O_p(G)\) is nontrivial. The main result in this paper shows that if this complex contains an acyclic \(2\)-dimensional \(G\)-subcomplex, then \(O_p(G)\) is nontrivial. As a consequence, the conjecture holds for groups of \(p\)-rank three. The authors also use their results to establish the conjecture for specific groups of \(p\)-rank three and four for which the main theorem in [\textit{M. Aschbacher} and \textit{S. D. Smith}, Ann. Math. (2) 137, No. 3, 473--529 (1993; Zbl 0782.20039)] does not apply. One of the examples at \(p=2\) is the semidirect product \((A_5 \times A_5) \rtimes \mathbb{Z}/2\), where \(\mathbb{Z}/2\) acts by conjugation by \((1,2)\) on each coordinate.