On intersections of nilpotent subgroups in finite groups with socle $L_2(2^m)\times L_2(2^n)$

In Theorem 1, it is proved for a finite group $G$ with socle $L_2(2^m)\times L_2(2^n)$ and nilpotent subgroups $A$ and $B$ that the condition $\min_G(A,B)\ne 1$ implies that $n=m=2$ and the subgroups $A$ and $B$ are $2$-groups. Here the subgroup $\min_G(A,B)$ is generated by smallest-order intersections of the form $A\cap B^g$, $g\in G$, and the subgroup $\mathrm{Min}_G(A,B)$ is generated by all intersections of the form $A\cap B^g$, $g\in G$, that are minimal with respect to inclusion. In Theorem 2, for a finite group $G$ with socle $A_5\times A_5$ and a Sylow 2-subgroup $S$, we give a description of the subgroups $\min_G(S,S)$ and $\mathrm{Min}_G(S,S)$. On the basis of Theorem 2, in Theorem 3 for a finite group $G$ with socle $A_5\times A_5$ we describe up to conjugation all pairs of nilpotent subgroups $(A,B)$ of $G$ for which $\min_G(A,B)\ne 1$.