ON GRAPH REPRESENTATION FOR CO-DIAGONAL GROUPS

For a co-diagonal Butler group G with co-representing graph, we examine the conditions under which pure subgroups and torsion-free quotients of G have quasi-co-representing graphs.

Note on the extensions of Butler groups

In terms of a new notion, called prebalancedness, those extensions of a Butler group by a Butler group which are again Butler groups are characterised. It is shown that the prebalanced extensions of torsion-free abelian groups form a divisible subgroup in the group of extensions.

Endomorphism rings of Butler groups

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.