Endomorphism rings of Butler groups

Abstract This note is devoted to the question of deciding whether or not a subring of a finite-dimensional algebra over the rationals, with additive group a Butler group, is the endomorphism ring of a Butler group (a Butler group is a pure subgroup of a finite direct sum of rank-1 torsion-free abelian groups). A complete answer is given for subrings of division algebras. Several applications are included.