Finite groups of the Lie type of small rank
Let \(G\) be a finite group. The Zassenhaus conjecture for integral group rings predicts that given two subgroups \(X\) and \(Y\) in the group of units of augmentation \(1\) of \(\mathbb{Z} G\), if the order of \(X\) and of \(Y\) equals the order of \(G\), then there is a unit \(a\) in \(\mathbb{Q} G\) so that \(aXa^{-1}=Y\). In case \(G\) has the property that any subgroup \(X\) as above is abstractly isomorphic to \(G\), then the conjecture is equivalent to the statement that any automorphism of \(\mathbb{Z} G\) preserving augmentation can be modified by a group automorphism of \(G\) so that the composite is the identity on the centre of \(\mathbb{Z} G\). Finite simple groups are such groups where one knows that the abstract isomorphism exists. False for solvable groups, the Zassenhaus conjecture remains open for finite simple groups. In the paper under review, the author proves the above Zassenhaus conjecture for all finite simple groups \(G\) of Lie type of rank \(1\) and of rank \(2\) which are not isomorphic to \(\text{PSU}_4(q^2)\) or \(\text{PSU}_5(q^2)\). Moreover, the author proves the conjecture for all finite simple groups with Abelian \(2\)-Sylow subgroups. The method of proof is to look at the action of an automorphism preserving the augmentation on the \(p\)-modular character table, on the ordinary character table and on the Brauer tree. Then, the proof is a case by case analysis of the different Dynkin diagrams and Brauer trees.
scientific article; zbMATH DE number 1344294