For a given cardinal
and
a torsion abelian group
of cardinality less than ,
we present, under some mild conditions (for example,
), boundedly endo-rigid
abelian group
of cardinality
with
.
Essentially, we give a complete characterization of such pairs
.
Among other things, we use a twofold version of the black box. We present an
application of the construction of boundedly endo-rigid abelian groups. Namely, we
turn to the existence problem of co-Hopfian abelian groups of a given size,
and present some new classes of them, mainly in the case of mixed abelian
groups. In particular, we give useful criteria to detect when a boundedly
endo-rigid abelian group is co-Hopfian and completely determine cardinals
for which there is a co-Hopfian abelian group of size
.
Keywords
black boxes, bounded endomorphisms, co-Hopfian groups,
endomorphism algebras, mixed abelian groups, $p$-groups,
set theoretical methods in algebra