# 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 (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 (or restricted direct product) of $\{f_{i}:G_{i}\to H_{i}\}_{i\in I}$.

Title direct products of homomorphisms DirectProductsOfHomomorphisms 2013-03-22 18:36:00 2013-03-22 18:36:00 joking (16130) joking (16130) 4 joking (16130) Definition msc 20A99