direct products of homomorphisms

Assume that {fi:GiHi}iI is a family of homomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath between groups. Then we can define the Cartesian productMathworldPlanetmath (or unrestricted direct product) of this family as a homomorphism


such that


for each giIGi and jI.

One can easily show that iIfi is a group homomorphism. Moreover it is clear that


so iIfi induces a homomorphism


which is a restrictionPlanetmathPlanetmathPlanetmath of iIfi to iIGi. This homomorphism is called the direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (or restricted direct product) of {fi:GiHi}iI.

Title direct products of homomorphisms
Canonical name DirectProductsOfHomomorphisms
Date of creation 2013-03-22 18:36:00
Last modified on 2013-03-22 18:36:00
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Definition
Classification msc 20A99