On the nilpotent complex of simple groups of Lie type. (Q1424049)

Let \(G\) be a finite group and \(t(G)\) be the number of connected components of the prime graph of \(G\), and \(Nl(G)\) be the partially ordered set of nilpotent subgroups of \(G\). In her previous paper [Commun. Algebra 23, No. 5, 1825-1836 (1995; Zbl 0831.20021)], the authoress proved that \(Nl(G)\) is not connected if and only if \(t(G)>1\) and determined the connected components of \(Nl(G)\), where \(G\) is soluble. In this paper, she determines the connected components of \(Nl(G)\) in the case when \(G\) is a simple non-Abelian group of Lie type. As a corollary, it is obtained that if \(t(G)>1\) in this case then the number of \(G\)-orbits of \(Nl(G)\) (\(G\) acts on \(Nl(G)\) by conjugacy) is exactly \(t(G)\).