Representations of Cyclic Groups in Rings of Integers, II

Abstract

Let Gk be a cyclic group of order k, and let ZGk denote its group ring over the ring Z of rational integers. We denote by n(ZGk) the number of non-isomorphic indecomposable left ZGk-modules having finite Z-bases. In 1938 Diederichsen [2] proved that n(ZG,) is finite for p a rational prime, and gave an incorrect proof that n(ZG4) is infinite. Recently Roiter [6] and Troy [8] independently showed that n(ZG4) = 9. The present authors [4] have shown more generally that n(ZGP2) is finite for all primes p. The aim of this paper is to establish the somewhat surprising result that n(ZGP3) is infinite for each prime p. Several interesting consequences of this result should be pointed out. To begin with, note that GP3 is a homomorphic image of Gp, for each r ? 3, and so each ZGP3-module may be viewed also as a ZGpr-module. Therefore n(ZGpr) _ n(ZGP3), which shows that n(ZGpr) is infinite for each r > 3. Next, let Hp denote (for the moment) a p-Sylow subgroup of an arbitrary finite group G. The authors have shown [4] that if n(ZHp) is infinite for some p, then also n(ZG) is infinite. They proved furthermore that n(ZHp) is necessarily infinite if Hp is non-cyclic. Combining this result with that of the present paper, it follows that n(ZHp) is finite if and only if Hp is cyclic of order p or p2. Thus, the possible groups G for which n(ZG) is finite are small in number; in fact, if n(ZG) is finite, then for each p dividing the order of G each p-Sylow subgroup of G must be cyclic of order p or p2. Whether the converse is true is as yet unknown.t Let Z* denote the ring of p-adic integers in the p-adic completion of the rational field. Define n(Z*Gpr) to be the number of non-isomorphic indecomposable left Z*Gpr-modules having finite Z*-bases. In their previous paper [4], the authors have shown that n(ZGpr) is finite if and only if n(Z*Gpr)is finite. It is therefore sufficienttoprovethatn(Z*GP3) is infinite. Section 1 contains some general remarks about extensions of one direct