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 $A\cap B^g\ne 1$ 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 $A\cap B^g\ne 1$ 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 $N\cap N^g=1$.