Direct Product of Group Homomorphisms is Homomorphism

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Theorem

Let $\left({G, \circ}\right), \left({H_1, *_1}\right)$ and $\left({H_2, *_2}\right)$ be groups.

Let $\left({H_1 \times H_2, *}\right)$ be the group direct product of $H_1$ and $H_2$.

Let $f_1: G \to H_1$ and $f_2: G \to H_2$ be group homomorphisms.

Let $f_1 \times f_2: g \to H_1 \times H_2$ be the direct product of $f_1$ and $f_2$.


Then $f_1 \times f_2$ is a group homomorphism.


Proof

The direct product of $f_1$ and $f_2$ $f_1 \times f_2: g \to H_1 \times H_2$ is defined as:

$\forall g \in G: \left({f_1 \times f_2}\right) \left({g}\right) = \left({f_1 \left({g}\right), f_2 \left({g}\right)}\right)$


From External Direct Product of Groups is Group, the group direct product $H_1 \times H_2$ is a group.


It remains to be shown that $f_1 \times f_2$ fulfils the morphism property.

Let $g, h \in G$.

Then:

\(\displaystyle \left({f_1 \times f_2}\right) \left({g}\right) * \left({f_1 \times f_2}\right) \left({h}\right)\) \(=\) \(\displaystyle \left({f_1 \left({g}\right), f_2 \left({g}\right)}\right) * \left({f_1 \left({h}\right), f_2 \left({h}\right)}\right)\) Definition of Direct Product of $f_1$ and $f_2$
\(\displaystyle \) \(=\) \(\displaystyle \left({f_1 \left({g}\right) *_1 f_1 \left({h}\right), f_2 \left({g}\right) *_2 f_2 \left({h}\right)}\right)\) Definition of Group Direct Product $\left({H_1 \times H_2, *}\right)$
\(\displaystyle \) \(=\) \(\displaystyle \left({f_1 \left({g \circ h}\right), f_2 \left({g \circ h}\right)}\right)\) $f_1$ and $f_2$ are Group Homomorphisms
\(\displaystyle \) \(=\) \(\displaystyle \left({f_1 \times f_2}\right) \left({g \circ h}\right)\) Definition of Direct Product of $f_1$ and $f_2$

$\blacksquare$


Sources