For a discussion of types and for basic definitions and notations see [ 7 1. In 1961 Beaumont and Pierce [4] posed the problem of finding necessary and sufficient conditions for a (necessarily finite or countable) set T of types to be realized as T = typeset G for some G of rank two. They presented a simple characterization for finite T: Let T be a finite set of types. Then T= typeset G for some G of rank two if and only if there exists a type t, in T such that inf(t, t’) = C, for all t # t’ in T. In general, if T is the typeset of a rank two group G, it is easy to see that there must exist a type f0 with inf(t, t’) = t, for all t # t’ in T (to is called the inner type of G). However, in 1965, Dubois [5] constructed a countable set of types T for which inf(t, t’) = type (2) for all t # t’ in T, but such that T is not the typeset of any rank two group. Some necessary and some sufficient conditions for T = typeset G, rank G = 2, were obtained by Koehler [9] in 1964, Dubois [S, 61 in 1965-1966, and Ito [8] in 1975. In 1978 Schultz [lo] introduced the term cotypeset for the set of types of all rank one factors of a group. This set of types has appeared with some regularity in the study of torsion-free Abelian groups [ 1,3, 10-15 1. In [ 101 Schultz claimed to have solved the problem of finding necessary and sufficient conditions on two sets of types T, T' such that T = typeset G, T' = cotypeset G for some rank two G. However, a counterexample to Schultz’ main theorem was given by Vinsonhaler and Wickless [ 131, where it was also shown that the “dual” to the Beaumont-Pierce condition was both 380 0021-8693/83 $3.00