Let V (ZG) be the normalized unit group of the integral group ring ZG of a finite group G. One of most interesting conjectures in the theory of integral group ring is the conjecture (ZC) of H. Zassenhaus [25], saying that every torsion unit u ∈ V (ZG) is conjugate to an element in G within the rational group algebra QG. For finite simple groups, the main tool of the investigation of the Zassenh...

It is shown that any torsion unit of the integral group ring ZG of a finite group G is rationally conjugate to a trivial unit if G = P o A with P a normal Sylow p-subgroup of G and A an abelian p′-group (thus confirming a conjecture of Zassenhaus for this particular class of groups). The proof is an application of a fundamental result of Weiss. It is also shown that the Zassenhaus conjecture ho...

We investigate the possible character values of torsion units of thenormalized unit group of the integral group ring of Mathieu sporadic groupM22. We confirm the Kimmerle conjecture on prime graphs for this group andspecify the partial augmentations for possible counterexamples to the strongerZassenhaus conjecture.

Zassenhaus Conjecture for torsion units states that every augmentation one torsion unit of the integral group ring of a finite group G is conjugate to an element of G in the units of the rational group algebra QG. This conjecture has been proved for nilpotent groups, metacyclic groups and some other families of groups. It has been also proved for some special groups. We prove the conjecture for...

Using the Luthar–Passi method and results of Hertweck, we study the long-standing conjecture of Zassenhaus for integral group rings of alternating groups An, n ≤ 8. As a consequence of our results, we confirm the Kimmerle’s conjecture about prime graphs for those groups.

We investigate the classical Zassenhaus conjecture for the unit group of the integral group ring of Mathieu simple group M 23 using the Luthar-Passi method. This work is a continuation of the research that we carried out for Mathieu groups M 11 and M 12. As a consequence, for this group we confirm Kimmerle's conjecture on prime graphs.