On Intersections of Certain Nilpotent Subgroups in Finite Groups

It is proved that, in any finite group $G$ with nilpotent subgroups $A$ and $B$ and the condition $A\cap B^g\unlhd\langle A,B^g\rangle$ for any $g$ in $G$, $\operatorname{Min}_G(A,B)$ is a subgroup of $F(G)$. This generalizes the author's theorem about intersections of Abelian subgroups in a finite group, since this holds, for example, for Hamiltonian subgroups $A$ and $B$ in $G$.