On $\sigma$-arithmetic graphs of finite groups

Let $G$ be a finite group and $\sigma$ a partition of the set of all? primes $\Bbb{P}$, that is, $\sigma =\{\sigma_i \mid i\in I \}$, where $\Bbb{P}=\bigcup_{i\in I} \sigma_i$ and $\sigma_i\cap \sigma_j= \emptyset $ for all $i\ne j$. If $n$ is an integer, we write $\sigma(n)=\{\sigma_i \mid \sigma_{i}\cap \pi (n)\ne \emptyset \}$ 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 $\sigma_j\in \sigma (N_{G}(H)/HC_{G}(H))$; (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 $G$.

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