Integral representations of dihedral groups of order 2𝑝

Abstract

Introduction. Information about the integral representations of finite groups has been obtained to varying extents. For Z the ring of rational integers and G the cyclic group of prime order, the ZG-modules were studied by Diederichsen [3] and Reiner [11], who showed that there were finitely many indecomposable ZG-modules and determined them completely. The finiteness of the number of indecomposables in the case where G is cyclic of order p2 was shown, for p = 2, by Troy [16] and for any p by Heller and Reiner [5] and by Knee [8], while Oppenheim [10] and Knee [8] established the finiteness of the number of indecomposables for G cyclic, of square free order. Heller and Reiner [5; 6] and Jones [7] established that the number of indecomposable ZG-modules is finite if and only if all p-Sylow subgroups of G are cyclic of order at most p2. Here, as well as throughout this paper, we shall mean by a ZG-module one which is finitely generated and Z-free. In this paper we shall classify all finitely-generated S-free SG-modules where G is the dihedral group of order 2p, p an odd prime, and S is Z or Z2p the semilocal ring formed by the intersection of Zp and Z2, respectively the rings of p-integral and 2-integral elements in s the rational field. Z2p = {r/seQ: (s,2p) = 1}. Taking 0 to be a primitive pth root of unity, we shall denote by K = Q(9) the cyclotomic field of degree p — 1 over s and by K0 = Q(9 + 9~l) the real subfield of K. R0 and R shall be the integral closures of S in K0 and K, respectively. Letting I) denote the group of automorphisms of K with fixed field K0, we may form A the twisted group ring of h with coefficients in R. §1 of this paper is devoted to a characterization of R-projective A-modules of finite R-rank. The results of this section are then applied in the second section to show that there are precisely 7/t + 3 nonisomorphic, indecomposable SGmodules where h is the ideal class number of R0. In §3 it is shown that although a Krull-Schmidt theorem is not obtainable for SG-modules, invariants may be obtained which determine an SG-module up to Z2pG-isomorphism. The final section deals with projective SG-modules. Here an isomorphism is established