On P-adic integral representations of a finite group.

It has been shown by F.E. Diederichsen that for integral representations of a finite group, the irreducible constituents in any complete reduction are note necessarily unique up to order and unimodular equivalence. In this same article, it is shown that for certain finite groups, such as the cyclic group of order 4, there are infinitely many classes of indecomposable representations under unimodular equivalence. A natural method for studying these problems of arithmetical representation theory would be the P-adic approach. As a first step in this direction, using the methods of Hensel and R. Brauer, we will show that the theory of representations of a finite group, over a ring of P-adic integers can always be brought back to the modular case, insofar as it is concerned with questions of unimodular equivalence, reduction and deomposition. More particularly, we will show that for any finite group, if P is a generator of the maximal ideal in the ring of P-adic intergers considered, and if Pko is the highest power of P dividing the order of the group, then unimodular equivalence may be considered module Pk, for any k > ko , while unimodular reduction and decomposition may be considered module Pk, for any k > 2 ko, without any loss of generality. As a corollary, we will show that if P does not divide the order of the group, then all questions of unimodular equivalence reduction and decomposition are completely equivalent to these same questions, module P. [...