decomposable homomorphisms and full families of groups

Let {Gi}iI,{Hi}iI be two families of groups (indexed with the same set I).

Definition. We will say that a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath


is decomposableMathworldPlanetmathPlanetmathPlanetmath if there exists a family of homomorphisms {fi:GiHi}iI such that


Remarks. For each jI and giIGi we will say that gGj if g(i)=0 for any ij. One can easily show that any homomorphism


is decomposable if and only if for any jI and any giIGi such that gGj we have f(g)Hj. This implies that if f is an isomorphismMathworldPlanetmathPlanetmathPlanetmath and f is decomposable, then each homomorphism in decomposition is an isomorphism and


Also it is worthy to note that compositionMathworldPlanetmathPlanetmath of two decomposable homomorphisms is also decomposable and


Definition. We will say that family of groups {Gi}iI is full if each homomorphism


is decomposable.

Remark. It is easy to see that if {Gi}iI is a full family of groups and I0I, then {Gi}iI0 is also a full family of groups.

Example. Let 𝒫={p|p is prime}. Then {p}p𝒫 is full. Indeed, let


be a group homomorphism. Then, for any q𝒫 and ap𝒫p such that aq we have that |a| divides q and thus |f(a)| divides q, so it is easy to see that f(a)q. Therefore (due to first remark) f is decomposable.

Counterexample. Let G1,G2 be two copies of . Then {G1,G2} is not full. Indeed, let


be a group homomorphism defined by


Now assume that f=f1f2. Then we have:


and so f2(0)=1. ContradictionMathworldPlanetmathPlanetmath, since group homomorphisms preserve neutral elements.

Title decomposable homomorphisms and full families of groups
Canonical name DecomposableHomomorphismsAndFullFamiliesOfGroups
Date of creation 2013-03-22 18:36:03
Last modified on 2013-03-22 18:36:03
Owner joking (16130)
Last modified by joking (16130)
Numerical id 7
Author joking (16130)
Entry type Definition
Classification msc 20A99