Arithmetic graphs and the products of finite groups

Abstract

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