On $\sigma$-arithmetic graphs of finite groups

Abstract

Let $G$ be a finite group and $\sigma$ a partition of the set of all? primes $\mathbb{P}$, that is, $\sigma =\{{\sigma }_i\mid i\in I\}$, where $\mathbb{P}={\bigcup }_{i\in I}{\sigma }_i$ and ${\sigma }_i\cap {\sigma }_j=\varnothing$ for all $i\ne j$. If $n$ is an integer, we write $\sigma (n)=\{{\sigma }_i\mid {\sigma }_i\cap \pi (n)\ne \varnothing \}$ and $\sigma (G)=\sigma (|G|)$. We call a graph $\Gamma$ with the set of all vertices $V(\Gamma )=\sigma (G)$ ($G\ne 1$) a $\sigma$-arithmetic graph of $G$, and we associate with $G\ne 1$ the following three directed $\sigma$-arithmetic graphs: (1) the $\sigma$-Hawkes graph ${\Gamma }_{H\sigma }(G)$ of $G$ is a $\sigma$-arithmetic graph of $G$ in which $({\sigma }_i,{\sigma }_j)\in E({\Gamma }_{H\sigma }(G))$ if ${\sigma }_j\in \sigma (G/F_{\{{\sigma }_i\}}(G))$; (2) the $\sigma$-Hall graph ${\Gamma }_{\sigma Hal}(G)$ of $G$ in which $({\sigma }_i,{\sigma }_j)\in E({\Gamma }_{\sigma Hal}(G))$ if for some Hall ${\sigma }_i$-subgroup $H$ of $G$ we have ${\sigma }_j\in \sigma (N_G(H)/H{C}_G(H))$; (3) the $\sigma$-Vasil'ev-Murashko graph ${\Gamma }_{{\mathfrak{N}}_{\sigma }}(G)$ of $G$ in which $({\sigma }_i,{\sigma }_j)\in E({\Gamma }_{{\mathfrak{N}}_{\sigma }}(G))$ if for some ${\mathfrak{N}}_{\sigma }$-critical subgroup $H$ of $G$ we have ${\sigma }_i\in \sigma (H)$ and ${\sigma }_j\in \sigma (H/F_{\{{\sigma }_i\}}(H))$. In this paper, we study the structure of $G$ depending on the properties of these three graphs of $G$.