Über ganzzahlige Permutationsdarstellungen einiger endlicher Gruppen (Q1844920)

The paper deals with the phenomenon that integral permutation representations of a finite group \(G\), which are rationally equivalent, somehow do resemble each other more closely than just arbitrary, rationally equivalent, integral representations of \(G\), e.g. their difference, taken in Swan's Grothendieck ring of integral representations of \(G\), is a torsion element, annihilated by the order and even by the Artin index of \(G\). lt is shown by exhibiting explicit isomorphisms that for certain finite groups including all abelian groups, this difference actually vanishes. The paper contains moreover a remark concerning integral relative Grothendieck rings over \(\mathbb Z\) (generalizing a correspondent result for such rings over the \(p\)-adic integers, derived by W. H. Gustafson), which is of importance for deriving induction theorems for those relative Grothendieck rings over arbitrary coefficient domains.