20.66 (2022)

A Schmidt $(p, q)$-group is a finite non-nilpotent group all of whose proper subgroups are nilpotent and whose Sylow $p$-subgroup is normal. The $N$-critical graph $\Gamma_{Nc}(G)$ of a finite group $G$ is a directed graph on the vertex set of all prime divisors of $|G|$ in which $(p, q)$ is an edge of $\Gamma_{Nc}(G)$ if and only if $G$ has a Schmidt $(p, q)$-subgroup.

Suppose that a finite group $G$ is such that $G = AB = AC = BC$, where $A, B, C$ are subgroups of $G$. Is
$$\Gamma_{Nc}(G) = \Gamma_{Nc}(A) \cup \Gamma_{Nc}(B) \cup \Gamma_{Nc}(C)?$$

This is true if $A, B, C$ are soluble.