direct products of homomorphisms

Assume that $\{f_{i}:G_{i}\to H_{i}\}_{i\in I}$ is a family of homomorphisms between groups. Then we can define the Cartesian product (or unrestricted direct product) of this family as a homomorphism

$\prod_{i\in I}f_{i}:\prod_{i\in I}G_{i}\to\prod_{i\in I}H_{i}$

such that

$\bigg{(}\prod_{i\in I}f_{i}\bigg{)}\big{(}g\big{)}(j)=f_{j}(g(j))$

for each $g\in\prod_{i\in I}G_{i}$ and $j\in I$ .

One can easily show that $\prod_{i\in I}f_{i}$ is a group homomorphism. Moreover it is clear that

$\bigg{(}\prod_{i\in I}f_{i}\bigg{)}\big{(}\bigoplus_{i\in I}G_{i}\big{)}% \subseteq\bigoplus_{i\in I}H_{i},$

so $\prod_{i\in I}f_{i}$ induces a homomorphism

$\bigoplus_{i\in I}f_{i}:\bigoplus_{i\in I}G_{i}\to\bigoplus_{i\in I}H_{i},$

which is a restriction of $\prod_{i\in I}f_{i}$ to $\bigoplus_{i\in I}G_{i}$ . This homomorphism is called the direct product (or restricted direct product) of $\{f_{i}:G_{i}\to H_{i}\}_{i\in I}$ .

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