A note on Hopfian and co-Hopfian abelian groups

The Hopfian exponent of an abelian group

“…Furthermore, as the group Z( p ∞ ) is not Hopfian, a Hopfian p-group is necessarily reduced. There is a further restriction on Hopfian p-groups that has no analogue in the torsion-free situation: a Hopfian p-group G is necessarily semi-standard, i.e., if B is a basic subgroup of G of the form B = ∞ n=1 B n , where each B n is the direct sum of copies of Z( p n ), then the subgroups B n are finite (or equivalently each finite Ulm invariant of G is finite)-see [9]. As a consequence Hopfian p-groups have cardinality at most the continuum.…”

“…It is well known (see e.g. [9]) that extensions of Hopfian torsion groups by torsion-free Hopfian groups are again Hopfian, so it is relatively easy to deal with splitting mixed groups: if G = T ⊕ F, where T is a torsion group with Hex(T ) = m and F is a torsion-free group with Hex(F) = n, then Hex(G) = min{m, n}; in particular, if m = n = ∞, then Hex(G) = ∞. The situation for non-splitting groups is, inevitably, more complex and results are somewhat sparser.…”

“…The situation for non-splitting groups is, inevitably, more complex and results are somewhat sparser. It is worth noticing that a mixed Hopfian group need not have a Hopfian torsion subgroup-see [9,Theorem 3.1].…”

“…We wish to focus on the mixed groups constructed in [9], so we recall some of the pertinent facts. Let T be a semi-standard unbounded reduced p-group and H a pure subgroup of rank 2 ℵ 0 of the group of p-adic integers such that H contains the subgroup Z p of integers localized at p and End(H ) = Z p ; note that it is immediate that H/Z p ∼ =…”

“…Fixing an isomorphism between H/Z p and A/T , we can form a group G by taking the pullback of A and H with kernels T and Z p ; note that T will be the torsion subgroup of G. Such a group G is shown to be Hopfian in [9] and the proof exploits the fact that if φ is an endomorphism of G, then there are integers m, n and t such that p t (mφ − n1 G ) = 0. It also follows from the construction-see [9, p. 134]-that multiplication by an integer q, with (q, p) = 1, is an automorphism of G. We shall call such groups mixed Corner Hopfian groups since the basic idea behind the construction in [9] is derived from an unpublished result of Corner ([U16] in [8]).…”