Admissible matrices as base changes of B (1)-groups: a realizing algorithm

In the study of Abelian groups a relevant role is played by realization of rings as endomorphism rings; this entails building from the ring a group on which the ring will act. We solve here a similar problem: given a ℚ-matrix M in the suitable class, build from it the Butler B (1)-groups for which M is a base change. The building procedure we give is an algorithm, easy to perform by hand in one-digit ranks.