Simple connectivity of \(p\)-group complexes
Let \(p\) be a prime and let \(G\) be a finite group. The commuting graph \(\Lambda_p(G)\) is the graph on the set of subgroups of \(G\) of order \(p\) whose edges are the pairs of commuting subgroups and the commuting complex is the clique complex \(K_p(G)=K(\Lambda_p(G))\), that is the simplicial complex whose simplices are the cliques of \(\Lambda_p(G)\). Also \(K_p(G)\) has the same homotopy type as the Brown complex and the Quillen complex for \(G\) at \(p\). This paper begins a systematic study of the following question: for which finite groups \(G\) and primes \(p\) is the commuting complex \(K_p(G)\) simply connected. Conjecture: Let the finite group \(G=AF^*(G)\) where \(A\) is an elementary abelian \(p\)-subgroup of rank at least 3 and \(F^*(G)\) is the direct product of the \(A\)-conjugates of a simple component \(L\) of \(G\) of order prime to \(p\). Then \(K_p(G)\) is simply connected. For a graph \(\Lambda\) and a vertex \(x\) of \(X\), let \(\Lambda(x)\) be the set of vertices of \(\Lambda\) adjacent to \(x\) and not \(x\). Theorem 1: Assume the conjecture and let \(G\) be a finite group, let \(p\) be a prime dividing \(|G|\) and set \(\Lambda=\Lambda_p(G)\) and set \(\overline{G}=G/{\mathcal O}_{p'}(G)\). Assume \(\Lambda(x)\) is connected for all \(x\in\Lambda\). Then exactly one of the following holds: (1) \(K_p(G)\) is simply connected. (2) \((\overline G)=\overline{G}_1\times\overline{G}_2\) where \(\overline{G}_i\) has a strongly \(p\)- embedded subgroup for \(i=1\) and 2. (3) \(\overline{G}=\overline{X}(\overline {G}_1\times\overline{G}_2)\) for some \(X\in\Lambda\), \(p=3,5\) and \(\overline{G}_1\cong L_2(8)\), \(\text{Sz}(32)\) resp., \(\overline{G}_2\) is a nonabelian simple group with a strongly \(p\)-embedded subgroup and \(X\) induces outer automorphisms on \(\overline{G}_i\) for \(i=1\) and 2. (4) \(\overline{G}\) is almost simple and \(K_p(\overline{G})\) and \(K_p(F^*(\overline{G}))\) are not simply connected. Theorem 2: Let \(G\) be a finite group and let the prime \(p\) divide \(|G|\). Assume that \({\mathcal O}_p(G)=1\), \(\Lambda=\Lambda_p(G)\) is connected and \(H_1(K_p(G))=0\). Then \(m_p(G)>2\) and \(\Lambda(x)\) is connected for each \(x\in\Lambda\). Theorem 3: Assume that \(G\) and \(L\) satisfy the hypotheses of the conjecture and that the conjecture holds in proper sections of \(G\). Then \(K_p(G)\) is simply connected if (1) \(L\) is of Lie type and Lie rank at least 2, (2) \(L\cong L_2(q)\) with \(q\) even, (3) \(L\) is an alternating group or (4) \(L\) is a Mathieu group. Assume the conjecture. Then Theorems 1 and 2 assert that, modulo a short list of exceptions, for any finite group \(G\), \(K_p(G)\) is simply connected if and only if \(m_p(G)>2\) and \(\Lambda(x)\) is connected for each \(x\in\Lambda_p(G)\). Moreover, if \(m_p(G)>2\), then \(\Lambda (x)\) is connected for all \(x\in\Lambda_p(G)\) unless \((G,p)\) is an exception listed in sections 7 and 8. Theorem 3 reduces a verification of the conjecture to the case where \(L\) is of Lie type and Lie rank 1 or \(L\) is a sporadic group not a Mathieu group. The proofs of these results use the theory developed by the author and Y. Segev for studying the simple connectivity of simplicial complexes in earlier work.
simple connectivity of simplicial complexes
scientific article; zbMATH DE number 446705
scientific article