Anisotropic groups of type \(A_n\) and the commuting graph of finite simple groups.
type: Item
instance of: scholarly article
title: Anisotropic groups of type \(A_n\) and the commuting graph of finite simple groups.
zbMATH Open document ID: 1058.20015
author: Yoav Segev
author: Gary M. Seitz
published in: Pacific Journal of Mathematics
publication date: 2002-10-22T00:00:00Z
review text: The well-known Margulis-Platonov conjecture describes the normal subgroup structure of an algebraic group \(G\) over a number field \(K\). The conjecture has been established for almost all isotropic groups and most anisotropic groups except for those of type \(A_n\). Inner forms of anisotropic groups \(G\) of type \(A_n\) have the form \(\text{SL}_{1,D}\), the reduced norm 1 group of a finite dimensional division algebra \(D\) over \(K\) [see \textit{V. P. Platonov} and \textit{A. S. Rapinchuk}, Algebraic groups and number theory, Pure Appl. Math. 139, Academic Press (1994; Zbl 0841.20046)]. In this case \textit{A. Potapchik} and \textit{A. Rapinchuk} [Proc. Indian Acad. Sci., Math. Sci. 106, No. 4, 329-368 (1996; Zbl 0879.20027)] showed that if \(\text{SL}_{1,D}\) fails to satisfy the conjecture then there exists a proper normal subgroup \(N\) of the multiplicative group \(D^*\) of \(D\) such that \(D^*/N\) is a nonabelian finite simple group. The commuting graph \(\Delta(H)\) of a finite group \(H\) is the graph whose vertex set is \(H\setminus Z(H)\) and whose edges are pairs \(\{h,g\}\subseteq H\setminus Z(H)\), such that \(h\not=g\) and \([h,g]\in Z(H)\); the diameter of \(\Delta(H)\) is denoted by \(\text{diam}(\Delta(H))\). If \(d\) is the distance function on \(\Delta(H)\), then \(\Delta(H)\) is balanced if there exists \(x,y\in\Delta(H)\) such that the distances \(d(x,y)\), \(d(x,xy)\), \(d(y,xy)\), \(d(x,x^{-1}y)\), \(d(y,x^{-1}y)\) are all larger than 3. The first author [Ann. Math. (2) 149, No. 1, 219-251 (1999; Zbl 0935.20009)] showed that if \(L\) is a nonabelian finite simple group and either \(\text{diam}(\Delta(H))>4\), or \(\Delta(H)\) is balanced, then \(L\) cannot be isomorphic to a quotient of \(D^*\). Consequently, the Margulis-Platonov conjecture for inner forms of anisotropic groups of type \(A_n\) is resolved by the following theorem, which is the main result of this paper: If \(L\) is a nonabelian finite simple group then either \(\text{di…
reviewed by: Anatoli Kondrat'ev
Mathematics Subject Classification ID: 20D05
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20G30
Mathematics Subject Classification ID: 11E57
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 16U60
zbMATH DE Number: 1818941
zbMATH Keywords: anisotropic algebraic groups over algebraic number fields
zbMATH Keywords: finite simple groups
zbMATH Keywords: commuting graphs
zbMATH Keywords: prime graphs
MaRDI profile type: Publication
full work available at URL: pjm.2002.202.125
OpenAlex ID: W1979281846
DOI: 10.2140/PJM.2002.202.125
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description: scientific article; zbMATH DE number 1818941