On type sets of torsion-free abelian groups of rank~2

Abstract

We prove existence of a torsion-free abelian group with a prescribed type-set. The existence of torsion-free abelian groups with prescribed type-set has been studied by Beaumont and Pierce [1] and Koehler [3], [4]. In this paper, we give an improved version of Theorem 7.10 in [1]i We refer the reader to [2] for the concepts of characteristic, type, and so forth. The type-set of a group G is denoted by T(G). As in [11, we say that a torsion-free abelian group G is completely anisotropic if r(x)W4 r(y) whenever x and y are linearly independent elements of G and r(x) denotes the type of the characteristic of x in G. We say that a set of characteristics Xo, X1, ... is relatively disjoint if Xi > Xo for all i > 1 and if, for all primes p and all i ; j, either xi(p) = X0(p) or Xj(p) = X0(p). A set of types 1ro, r1,... I is relatively disjoint if it can be represented by a relatively disjoint set of characteristics. (This is stronger than merely requiring that r= ri nl r. for i; j.) 0 I Theorem I. Let T = tr, r,.. . be a countably infinite relatively disjoint set of types. Then there exists a completely anisotropic rank two torsion-free abelian group A such that T(A) = T. Proof. It clearly suffices to prove the case r0 = T(Z), where Z is the group of integers. We may then choose characteristics Xi representing the ri in such a way that for all primes P, xo(p) = 0 and for all i; j, either Xi(p) = or Xj(p) = 0. Let 7rn = 'PlXn(P) > 0} for n > 1, and let Pn = min 77 Thus 7ri. n ir. = 0 for i j. We may suppose that the Xn have been numbered in such a way as to satisfy the following: (*) For all n, Pn < Pn+1' Received by the editors November 1, 1973 and, in revised form, January 15, 1974. AMS (MOS) subject classifications (1970). Primary 20K15.