The N-prime graph and the Subgroup Isomorphism Problem

TL;DR

This paper introduces the N-prime graph, a new refinement of the Gruenberg-Kegel graph, to analyze subgroup structures and unit groups in integral group rings, providing new insights into the Subgroup Isomorphism Problem.

Contribution

It defines the N-prime graph and proves its equivalence for units and the group in certain classes, advancing understanding of subgroup structures and the Subgroup Isomorphism Problem.

Findings

The N-prime graph of units matches that of the group for finite solvable groups.

The N-prime graph equality holds for almost simple groups with specific socles.

Stronger results for solvable groups relate Frobenius subgroups in units to those in the group.

Abstract

We introduce a directed graph related to a group $G$, which we call the N-prime graph $\Gamma_{\rm{N}}(G)$ of $G$ and which is a refinement of the classical Gruenberg-Kegel graph. The vertices of $\Gamma_{\rm{N}}(G)$ 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 $\Gamma_{\rm{N}}(V(\mathbb{Z} G))=\Gamma_{\rm{N}}(G)$ holds for any finite group $G$. We also prove that $\Gamma_{\rm{N}}(V(\mathbb{Z} G))=\Gamma_{\rm{N}}(G)$ 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 $T$.