Seminario Teoría de Grupos
The N-prime graph and the subgroup isomorphism problem
Ponente: Marco Vergani (Università degli studi di Firenze)Fecha: jueves 30 de abril de 2026 - 11:30Lugar: Aula Naranja, ICMAT
Resumen:
The study of the unit group of an integral group ring has long been driven by fundamental questions, such as the Zassenhaus Conjecture, Subgroup Isomorphism Problem and the Prime Graph Question. We introduce a directed graph related to a group G, which we call the N-prime graph of G and which is a refinement of the classical Gruenberg-Kegel graph. Generalizing some known results about the Gruenberg-Kegel graph, we proved that the group of the units with augmentation 1 in the integral group ring has the same N-prime graph as G if G is a finite solvable group, and we reduced to almost simple groups the problem of whether the N-prime graph of G coincides the N-prime graph of the group of normalized units of the integral group ring of G holds for any finite group G. Finally, for G solvable we obtain some stronger results which give a contribution to the Subgroup Isomorphism Problem. More precisely, we prove that if the group of normalized units of the integral group ring of 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.
