full families of Hopfian (co-Hopfian) groups

PropositionPlanetmathPlanetmathPlanetmath. Let {Gi}iI be a full family of groups. Then each Gi is Hopfian (co-Hopfian) if and only if iIGi is Hopfian (co-Hopfian).

Proof. ,,” Let


be a surjectivePlanetmathPlanetmath (injectivePlanetmathPlanetmath) homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. Since {Gi}iI is full, then there exists family of homomorphisms {fi:GiGi}iI such that


Of course since f is surjective (injective), then each fi is surjective (injective). Thus each fi is an isomorphismMathworldPlanetmathPlanetmath, because each Gi is Hopfian (co-Hopfian). Therefore f is an isomorphism, because


,,” Fix jI and assume that fj:GjGj is a surjective (injective) homomorphism. For iI such that ij define fi:GiGi to be any automorphism of Gi. Then


is a surjective (injective) group homomorphism. Since iIGi is Hopfian (co-Hopfian) then iIfi is an isomorphism. Thus each fi is an isomorphism. In particular fj is an isomorphism, which completesPlanetmathPlanetmathPlanetmathPlanetmath the proof.

Example. Let 𝒫={p|p is prime} and 𝒫0 be any subset of 𝒫. Then


is both Hopfian and co-Hopfian.

Proof. It is easy to see that {p}p𝒫 is full, so {p}p𝒫0 is also full. Moreover for any pP0 the group p is finite, so both Hopfian and co-Hopfian. Therefore (due to proposition)


is both Hopfian and co-Hopfian.

Title full families of Hopfian (co-Hopfian) groups
Canonical name FullFamiliesOfHopfiancoHopfianGroups
Date of creation 2013-03-22 18:36:05
Last modified on 2013-03-22 18:36:05
Owner joking (16130)
Last modified by joking (16130)
Numerical id 7
Author joking (16130)
Entry type Theorem
Classification msc 20A99