The N-Prime Graph Question is equivalent to the Prime Graph Question

Abstract

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 $A(\Delta)$ for the arc set of a directed graph $\Delta$, $E(\Delta)$ 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 \[ A\bigl(\Gamma_{\mathrm N}(V(\mathbb ZG))\bigr) = A\bigl(\Gamma_{\mathrm N}(G)\bigr) \cup \operatorname{Sym}\!\bigl( E(\Gamma_{\mathrm{GK}}(V(\mathbb ZG))) \bigr). \] Consequently, the $N$-Prime Graph Question has an affirmative answer for $G$ if and only if the Prime Graph Question does.