On Quillen's conjecture for the \(p\)-groups complex
If \(G\) is a finite group and \(p\) a prime, then \({\mathcal S}_ p(G)\) is the simplicial complex consisting of all nontrivial \(p\)-subgroups of \(G\). \textit{D. Quillen} [Adv. Math. 28, 101-128 (1978; Zbl 0388.55007)] proved that \({\mathcal S}_ p(G)\) is homotopy equivalent to its subcomplex \({\mathcal A}_ p(G)\) consisting of the nontrivial elementary \(p\)-subgroups and that \({\mathcal A}_ p(G)\) is contractible provided \(O_ p(G)\neq 1\). He conjectured also that the converse statement should hold: if \(O_ p(G) = 1\), then \(\widetilde{\text{H}}_*({\mathcal A}_ p(G))\neq 0\). The main result of the paper under review is the following Theorem: Quillen's conjecture holds when \(p>5\) and \(G\) has no components \(U_ n(q)\) with \(q\equiv-1\pmod p\) and \(p\) odd.
scientific article; zbMATH DE number 205463