The prime graph conjecture for integral group rings of some alternatings groups

We investigate the classical Zassenhaus Conjecture (ZC) for integral group rings of alternating groups A9 and A10. Even the question (ZC) remains open as no counterexample is known up to date, it been confirmed for special types of groups such as nilpotent groups by Roggenkamp, Scot and Weiss. However, a new method based on the partial augmentation of torsion units been established by Luthar and Passi to confirm the (ZC) for A5. Later a weaker version of (ZC) was proposed in 2007, we call it the Prime Graph Conjecture (PGC) about the Gruenberg-Kegel (prime) graph of the group of all normalized units of the integral group ring of a finite group. Recently, the (PGC) has a positive answer for solvable groups, Frobenius groups and several simple groups. Here, as a consequence of our results, we confirm the (PGC) for integral group rings of alternating groups An for all n<11.