N-素图问题等价于素图问题
The N-Prime Graph Question is equivalent to the Prime Graph Question
AI总结：
该研究证明有限群$G$的N-素图问题与素图问题等价，给出了整群环正规化单位群相关的弧集公式，为群论中两类核心问题的关联提供了关键结论。
AI中文摘要：
设$G$为有限群，$V(\boldsymbol{Z}G)$为其整群环的正规化单位群。我们证明，$V(\boldsymbol{Z}G)$的每一条N-素弧要么已在$G$中出现，要么存在不同素数阶的交换见证元。记$A(\triangle)$为有向图$\triangle$的弧集，$E(\triangle)$为无向图$\triangle$的边集，$\text{Sym}(E)$为$E$中边的两种定向，则该结论等价于精确公式：$A\bigl(\boldsymbol{\text{Γ}}_{\text{N}}(V(\boldsymbol{Z}G))\bigr) = A\bigl(\boldsymbol{\text{Γ}}_{\text{N}}(G)\bigr) \bigcup \text{Sym}\bigl(E(\boldsymbol{\text{Γ}}_{\text{GK}}(V(\boldsymbol{Z}G)))\bigr)$。因此，$G$的N-素图问题有肯定答案当且仅当素图问题有肯定答案。
英文摘要：
Let $G$ be a finite group and let $V(\mathbb ZG)$ be the group of normalized units of its integral group ring. We prove that every $N$-prime arc of $V(\mathbb ZG)$ either already occurs in $G$ or admits commuting witnesses of distinct prime orders. Writing $A(Δ)$ for the arc set of a directed graph $Δ$, $E(Δ)$ for the edge set of an undirected graph, and $\operatorname{Sym}(E)$ for the two orientations of the edges in $E$, this is equivalent to the exact formula \[ A\bigl(Γ_{\mathrm N}(V(\mathbb ZG))\bigr) = A\bigl(Γ_{\mathrm N}(G)\bigr) \cup \operatorname{Sym}\!\bigl( E(Γ_{\mathrm{GK}}(V(\mathbb ZG))) \bigr). \] Consequently, the $N$-Prime Graph Question has an affirmative answer for $G$ if and only if the Prime Graph Question does.
