On Intersections of Nilpotent Subgroups in Finite Groups with Simple Socle from the “Atlas of Finite Groups”

Abstract

Earlier, the author described up to conjugacy all pairs \((A,B)\) of nilpotent subgroups of a finite group \(G\) with socle \(L_{2}(q)\) for which \(A\cap B^{g}\neq 1\) for any element of \(G\). A similar description was obtained by the author later for primary subgroups \(A\) and \(B\) of a finite group \(G\) with socle \(L_{n}(2^{m})\). In this paper, we describe up to conjugacy all pairs \((A,B)\) of nilpotent subgroups of a finite group \(G\) with simple socle from the “Atlas of Finite Groups” for which \(A\cap B^{g}\neq 1\) for any element \(g\) of \(G\). The results obtained in the considered cases confirm the hypothesis (Problem 15.40 from the “Kourovka Notebook”) that a finite simple nonabelian group \(G\) for any nilpotent subgroups \(N\) contains an element \(g\) such that \(N\cap N^{g}=1\).