PreprintCC BY 4.0arXiv
The N-Prime Graph Question is equivalent to the Prime Graph Question
Abstract
Let G be a finite group and let V(ℤG) be the group of normalized units of its integral group ring. We prove that every N-prime arc of V(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 Sym(E) for the two orientations of the edges in E, this is equivalent to the exact formula A(ΓN(V(ℤG))) = A(ΓN(G)) ∪ Sym ( E(ΓGK(V(ℤG))) ). Consequently, the N-Prime Graph Question has an affirmative answer for G if and only if the Prime Graph Question does.
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BibTeX
@misc{verbeken2026nprime,
title = {{The N-Prime Graph Question is equivalent to the Prime Graph Question}},
author = {Brecht F. Verbeken},
howpublished = {arXiv},
year = {2026},
month = jul,
doi = {10.48550/arxiv.2607.29105},
eprint = {2607.29105},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.29105},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The N-Prime Graph Question is equivalent to the Prime Graph Question
AU - Brecht F. Verbeken
PY - 2026
DA - 2026/07/31
PB - arXiv
DO - 10.48550/arxiv.2607.29105
UR - https://arxiv.org/abs/2607.29105
AB - Let G be a finite group and let V(ℤG) be the group of normalized units of its integral group ring. We prove that every N-prime arc of V(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 Sym(E) for the two orientations of the edges in E, this is equivalent to the exact formula A(Γ_N(V(ℤG))) = A(Γ_N(G)) ∪ Sym ( E(Γ_GK(V(ℤG))) ). Consequently, the N-Prime Graph Question has an affirmative answer for G if and only if the Prime Graph Question does.
ER -