Some Indecomposable Group Representations

In [1], Heller and Reiner have shown that any p-group G has infinitely many inequivalent, indecomposable rational integral representations, unless G is cyclic of order p or p2. In the present paper, we shall derive these results from a more general structural theorem in ? 2 below. This theorem states that, if 0 is any order in any algebraic number field K such that the p-group G has at least four distinct K-irreducible representations, then the group ring O[G] has infinitely many inequivalent, indecomposable torsion-free modules. From the available evidence, it seems that the converse to this statement ought to hold, i.e., that if G has three or fewer distinct K-irreducible representations, then O[G] has only finitely many distinct indecomposable torsion-free modules. Warning. We remark here once for all that any ring in this paper comes provided with an identity; any module is a left module; and the ring identity acts as the identity transformation of the module.