Tensor products and a conjecture of Zassenhaus (Q1805512)

It is shown that a certain conjecture of Zassenhaus is valid for all minimal simple groups and all simple Zassenhaus groups. For a group \(G\) of this type, two subgroups of the group of units of \(\mathbb{Z} G\) with augmentation 1 which have the same order as \(G\) are conjugate by a unit of \(\mathbb{Q} G\). In order to establish these results, the action of augmentation preserving ring automorphisms of \(\mathbb{Z} G\) on tensor products of characters is determined. Furthermore generic ordinary and modular character tables are examined.