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 $q\in \pi (G/O_{p^{\prime },p}(G))$. 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 $q\in \pi (N_G(P)/P{C}_G(P))$ 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.