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

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 \(N\cap N^{g}=1\).