Arithmetic graphs and factorized finite groups

The Hawkes graph $\Gamma_H(G)$ of a group $G$ is the directed graph with vertex set $\pi(G)$ that has an edge $(p, q)$ whenever $q\in\pi(G/O_{p',p}(G))$. The Sylow graph $\Gamma_s(G)$ of a group $G$ is the directed graph with vertex set $\pi(G)$ that has an edge $(p, q)$ whenever $q \in\pi(N_G(P)/PC_G(P))$ for some Sylow $p$-subgroup $P$ of $G$. The $N$-critical graph $\Gamma_{Nc}(G)$ of a group $G$ is the directed graph with vertex set $\pi(G)$ that has an edge $(p, q)$ whenever $G$ contains a Schmidt $(p, q)$-subgroup, i.e., a Schmidt $\{p, q\}$-subgroup with a normal Sylow $p$-subgroup. The paper studies the Hawkes, Sylow, and $N$-critical graphs of products of totally permutable, mutually permutable, and $\mathfrak{N}$-connected subgroups.