论文标题
有限组的$σ$ -Arithmetic图
On $σ$-arithmetic graphs of finite groups
论文作者
论文摘要
让$ g $为有限的群体，而$σ$ a的分区？ Primes $ \ bbb {p} $，也就是说，$σ= \ {σ_i\ mid i \ in I \} $，其中$ \ bbb {p} = \ bigCup_ {i \ in I}σ_i$ and un和$σ_i\capσ_jj= $ j $ y $如果$ n $是整数，我们写$σ（n）= \ {σ_i\midσ_{i} \capπ（n）\ ne \ ne \ emptyset \} $和$σ（g）=σ（| g |）$。我们将所有顶点$ v（γ）=σ（g）$（$ g \ ne 1 $）的集合$γ$称为图形$γ$，$ g $的$σ$ -Arithmetic图，我们与$ g \ ne 1 $ 1 $相关联，以下三个指导$σ$σ$ - arithmetic graphs：（ $σ$ -ARITHMETIC图$ g $中的$（σ_i，σ_j）\在e（γ_{Hσ}（g））中$如果$σ_j\ inσ（g/f _ {\ {\ {σ_i\}}}（g）（g）（g）（g）（g）（g））$; （2）$σ$ -HALL图$γ_{σhal}（g）$ g $的$（c $σ_i$ -cument $ -subgroup $ -subgroup $ h $ g $ $ g $ n o $ g $ $ g $ n e（γ_i{σhal}（g））$（σ_i，σ_j）\ in E（γ_{σhal}（g））$ σ（n_ {g}（h）/hc_ {g}（h））$; （3）$σ$ -VASIL'EV-MURASHKO图$γ_ {{{\ Mathfrak {n}_σ}}}}}}（g）$ g $ of $ g $，其中$（σ_i，σ_j） $ {\ mathfrak {n} _ {σ}} $ - 关键子组$ h $的$ g $我们有$σ_i\ inσ（h）$和$σ_j\ inσ（h/f _ {\ {\ {σ_i\}}（h））$。在本文中，我们研究了$ g $的结构，具体取决于这三个图的$ g $的属性。
Let $G$ be a finite group and $σ$ a partition of the set of all? primes $\Bbb{P}$, that is, $σ=\{σ_i \mid i\in I \}$, where $\Bbb{P}=\bigcup_{i\in I} σ_i$ and $σ_i\cap σ_j= \emptyset $ for all $i\ne j$. If $n$ is an integer, we write $σ(n)=\{σ_i \mid σ_{i}\cap π(n)\ne \emptyset \}$ and $σ(G)=σ(|G|)$. We call a graph $Γ$ with the set of all vertices $V(Γ)=σ(G)$ ($G\ne 1$) a $σ$-arithmetic graph of $G$, and we associate with $G\ne 1$ the following three directed $σ$-arithmetic graphs: (1) the $σ$-Hawkes graph $Γ_{Hσ}(G)$ of $G$ is a $σ$-arithmetic graph of $G$ in which $(σ_i, σ_j)\in E(Γ_{Hσ}(G))$ if $σ_j\in σ(G/F_{\{σ_i\}}(G))$; (2) the $σ$-Hall graph $Γ_{σHal}(G)$ of $G$ in which $(σ_i, σ_j)\in E(Γ_{σHal}(G))$ if for some Hall $σ_i$-subgroup $H$ of $G$ we have $σ_j\in σ(N_{G}(H)/HC_{G}(H))$; (3) the $σ$-Vasil'ev-Murashko graph $Γ_{{\mathfrak{N}_σ}}(G)$ of $G$ in which $(σ_i, σ_j)\in E(Γ_{{\mathfrak{N}_σ}}(G))$ if for some ${\mathfrak{N}_{σ}}$-critical subgroup $H$ of $G$ we have $σ_i \in σ(H)$ and $σ_j\in σ(H/F_{\{σ_i\}}(H))$. In this paper, we study the structure of $G$ depending on the properties of these three graphs of $G$.
