3 December 2025 — Padova

N-Prime graph and units of integral group rings

Ángel Del Río, University of Murcia

Abstract

The N-prime graph of a group \(G\) is the directed graph whose vertices are the primes occurring as the order of an element of \(G\), and the arrows are of the form \(p\to q\) for \(p\) and \(q\) different vertices such that \(G\) contains an element of order \(p\) whose normalizer in \(G\) has an element of order \(q\).

We will present some recent results, obtained in cooperation with E. Pacifici and M. Vergani, on the following question: Does a finite group \(G\) has the same N-prime graph than the group \(V(\mathbb{Z} G)\) of normalized units of its integral group ring? We will also present connections of these results with the following question, know as the Subgroups Isomorphism Problem: For which finite groups \(T\) and \(G\), is it true that if \(V(\mathbb{Z} G)\) contains a subgroup isomorphic to \(T,\) then so does \(G\)?