The N-prime graph and the Subgroup Isomorphism Problem

Abstract

We introduce a directed graph related to a group $G$, which we call the N-prime graph $\hat{I}{“}_{\mathrm{N}}(G)$ of $G$ and which is a refinement of the classical Gruenberg-Kegel graph. The vertices of $\hat{I}{“}_{\mathrm{N}}(G)$ are the primes $p$ such that $G$ has an element of order $p$, and, for distinct vertices $p$ and $q$, the arc $q\to p$ 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 $\hat{I}{“}_{\mathrm{N}}(V(\mathbb{Z}G))=\hat{I}{“}_{\mathrm{N}}(G)$ holds for any finite group $G$. We also prove that $\hat{I}{“}_{\mathrm{N}}(V(\mathbb{Z}G))=\hat{I}{“}_{\mathrm{N}}(G)$ if $G$ is almost simple with socle either an alternating group, or $\mathrm{PSL}(r^f)$ 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 $T$.