On a conjecture of Quillen and a lemma of Robinson
By definition, if \(X\) is a finite group, then \(X\) is said to be hyperelementary if \(O^ q(X)\) is cyclic for some prime \(q\). If \(X\) is a subgroup of the finite group \(G\) and \(p\) is a prime, then \(\text И_ G(X,p)\) denotes the set of all \(p\)-subgroups of \(G\) normalized by \(X\) and \({\mathcal S}_ p(G)\) denotes the simplicial complex of the partially ordered (under inclusion) set of nontrivial \(p\)-subgroups of \(G\) in Quillen's conjecture. Also the Robinson properties are: (R1) All nilpotent \(p'\)-subgroups \(X\) of \(F^*(G)\) satisfy \(\text И_ G(X,p)\neq 1\). (R2) All hyperelementary nilpotent \(p'\)-subgroups \(X\) of \(F^*(G)\) satisfy \(\text И_ G(X,p)\neq 1\). The first of three main results of this paper is: Theorem 1. Let \(G\) be a finite group and \(p\) a prime. If \((G,p)\) satisfies the Robinson property (R1), then (a) \(O_ p(G)\neq 1\); or (b) \(p=2\), \(G\) has a component \(L\cong L_ 3(4)\) and \(4\mid Out_ G(L)\). - Theorem 1 follows readily from: Theorem 2. Let \(G\) be an almost simple finite group (i.e., \(F^*(G)\) is simple) and \(p\) a prime. Then the following are equivalent: (i) \((G,p)\) have the Robinson property (R2); and (ii) \(p=2\), \(F^*(G)\cong L_ 3(4)\) and \(4\mid|G/F^*(G)|\). - The proof of Theorem 2 invokes the classification of finite simple groups. Theorem 2 and Robinson's Lemma are used to prove: Theorem 3. Let \(G\) be an almost simple finite group and \(p\) a prime. Then \({\mathcal S}_ p(G)\) is not contractible. The authors remark that Theorem 3 could be strengthened to allow \(G/O_{p'}(G)\) to be the direct product of almost simple groups.
scientific article; zbMATH DE number 4191984