Multiplicative Invariants and the Finite Co-Hopfian Property

A group is said to be, finitely co-Hopfian when it contains no proper subgroup of finite index isomorphic to itself. It is known that irreducible lattices in semisimple Lie groups are finitely co-Hopfian. However, it is not clear, and does not appear to be known, whether this property is preserved under direct product. We consider a strengthening of the finite co-Hopfian condition, namely the existence of a non-zero multiplicative invariant, and show that, under mild restrictions, this property is closed with respect to finite direct products. Since it is also closed with respect to commensurability, it follows that lattices in linear semisimple groups of general type are finitely co-Hopfian

On hereditarily and super R-Hopfian and L-co-Hopfian abelian groups

An Abelian group is said to be R-Hopfian [L-co-Hopfian] if every surjective [injective] endomorphism has a right [left] inverse. An Abelian group G is said to be hereditarily R-Hopfian [hereditarily L-co-Hopfian] if each subgroup of G is R-Hopfian [L-co-Hopfian]; similarly G is super R-Hopfian [super L-co-Hopfian] if each homomorphic image of G is R-Hopfian [L-co-Hopfian]. The various classes of hereditarily and super R-Hopfian and L-co-Hopfian groups are studied and necessary conditions for groups to have these properties are derived; in several, but not all, cases, sufficient conditions are also obtained

On Super and Hereditarily Hopfian and co-Hopfian Abelian Groups

The notions of Hopfian and co-Hopfian groups have been of interest for some time. In this present work we characterize the more restricted classes of hereditarily Hopfian (co-Hopfian) and super Hopfian (co-Hopfian) groups in the case where the groups are Abelian

On the Existence of Uncountable Hopfian and co-Hopfian Abelian Groups

We deal with the problem of existence of uncountable co-Hopfian abelian groups and (absolute) Hopfian abelian groups. Firstly, we prove that there are no co-Hopfian reduced abelian groups $G$ of size $< \mathfrak{p}$ with infinite $\mathrm{Tor}_p(G)$, and that in particular there are no infinite reduced abelian $p$-groups of size $< \mathfrak{p}$. Secondly, we prove that if $2^{\aleph_0} < \lambda < \lambda^{\aleph_0}$, and $G$ is abelian of size $\lambda$, then $G$ is not co-Hopfian. Finally, we prove that for every cardinal $\lambda$ there is a torsion-free abelian group $G$ of size $\lambda$ which is absolutely Hopfian, i.e., $G$ is Hopfian and $G$ remains Hopfian in every forcing extensions of the universe

Hopfian and co-hopfian subsemigroups and extensions

This paper investigates the preservation of hopficity and co-hopficity on passing to finite-index subsemigroups and extensions. It was already known that hopficity is not preserved on passing to finite Rees index subsemigroups, even in the finitely generated case. We give a stronger example to show that it is not preserved even in the finitely presented case. It was also known that hopficity is not preserved in general on passing to finite Rees index extensions, but that it is preserved in the finitely generated case. We show that, in contrast, hopficity is not preserved on passing to finite Green index extensions, even within the class of finitely presented semigroups. Turning to co-hopficity, we prove that within the class of finitely generated semigroups, co-hopficity is preserved on passing to finite Rees index extensions, but is not preserved on passing to finite Rees index subsemigroups, even in the finitely presented case. Finally, by linking co-hopficity for graphs to co-hopficity for semigroups, we show that without the hypothesis of finite generation, co-hopficity is not preserved on passing to finite Rees index extensions.Comment: 15 pages; 3 figures. Revision to fix minor errors and add summary table