Simple connectivity of \(p\)-group complexes
type: Item
instance of: scholarly article
title: Simple connectivity of \(p\)-group complexes
zbMATH Open document ID: 0829.20031
DOI: 10.1007/BF02808107
author: Michael Aschbacher
published in: Israel Journal of Mathematics
publication date: 1996-01-15T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 55U10
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 55P10
zbMATH DE Number: 446705
zbMATH Keywords: subgroups of order \(p\)
zbMATH Keywords: connected commuting complex
zbMATH Keywords: groups of Lie type
zbMATH Keywords: commuting graph
zbMATH Keywords: commuting subgroups
zbMATH Keywords: commuting complex
zbMATH Keywords: homotopy type
zbMATH Keywords: Brown complex
zbMATH Keywords: Quillen complex
zbMATH Keywords: finite groups
zbMATH Keywords: elementary Abelian \(p\)- subgroups
zbMATH Keywords: rank
zbMATH Keywords: simple components
zbMATH Keywords: strongly \(p\)-embedded subgroups
zbMATH Keywords: alternating groups
zbMATH Keywords: Mathieu groups
zbMATH Keywords: sporadic groups
zbMATH Keywords: simple connectivity of simplicial complexes
reviewed by: Morton E. Harris
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 446705