The N-prime graph and the Subgroup Isomorphism Problem

TL;DR

This work introduces the N-prime graph $\Gamma_{\rm{N}}(G)$, a directed refinement of the Gruenberg-Kegel graph that encodes normalization relations between $p$-elements and $q$-elements in a finite group. It proves a foundational reduction: if NPQ holds for every almost simple image of $G$, then NPQ holds for $G$, and as a consequence NPQ is valid for all finite solvable groups. The authors establish NPQ for almost simple groups with socle $A_n$ or $\mathrm{PSL}_2(r^f)$ ($r$ prime, $f\in\{1,2\}$), leveraging a blend of character theory and GK-graph correspondences. Finally, they contribute to the Subgroup Isomorphism Problem by showing that for solvable $G$, the presence of a Frobenius subgroup $C_p\rtimes C_{q^k}$ in $V({\mathbb Z}G)$ guarantees a corresponding subgroup in $G$, with extensions to metacyclic structures and the case $G'$ cyclic. Overall, the paper strengthens NPQ by reducing it to almost simple images and provides new positive results for key group families, enriching the understanding of integral group rings and subgroup realizability.

Abstract

We introduce a directed graph related to a group $G$, which we call the N-prime graph $Γ_{\rm{N}}(G)$ of $G$ and which is a refinement of the classical Gruenberg-Kegel graph. The vertices of $Γ_{\rm{N}}(G)$ are the primes $p$ such that $G$ has an element of order $p$, and, for distinct vertices $p$ and $q$, the arc $q\rightarrow p$ is in the graph if and only if $G$ has a subgroup of order $p$ whose normalizer in $G$ has an element of order $q$. Generalizing some known results about the Gruenberg-Kegel graph, we prove that the group $V(\mathbb{Z} G)$ of the units with augmentation $1$ in the integral group ring $\mathbb{Z} G$ has the same N-prime graph as $G$ if $G$ is a finite solvable group, and we reduce to almost simple groups the problem of whether $Γ_{\rm{N}}(V(\mathbb{Z} G))=Γ_{\rm{N}}(G)$ holds for any finite group $G$. We also prove that $Γ_{\rm{N}}(V(\mathbb{Z} G))=Γ_{\rm{N}}(G)$ if $G$ is almost simple with socle either an alternating group, or ${\rm{PSL}}(r^f)$ with $r$ prime and $f\le 2$. Finally, for $G$ solvable we obtain some stronger results which give a contribution to the Subgroup Isomorphism Problem. More precisely, we prove that if $V(\mathbb{Z} G)$ contains a Frobenius subgroup $T$ with kernel of prime order and complement of prime power order, then $G$ contains a subgroup isomorphic to $T$.

Theorems & Definitions (29)

• Theorem A
• Corollary B
• Theorem C
• Theorem D
• Theorem E
• Remark 3.1
• Lemma 3.2
• proof
• Proposition 3.3
• proof