# Direct Product of Group Homomorphisms is Homomorphism

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## Theorem

Let $\struct {G, \circ}, \struct {H_1, *_1}$ and $\struct {H_2, *_2}$ be groups.

Let $\struct {H_1 \times H_2, *}$ 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:

- $\forall g \in G: \map {\paren {f_1 \times f_2} } g = \tuple {\map {f_1} g, \map {f_2} g}$

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:

\(\displaystyle \) | \(\) | \(\displaystyle \map {\paren {f_1 \times f_2} } g * \map {\paren {f_1 \times f_2} } h\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \tuple {\map {f_1} g, \map {f_2} g} * \tuple {\map {f_1} h, \map {f_2} h}\) | Definition of Direct Product of $f_1$ and $f_2$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \tuple {\map {f_1} g *_1 \map {f_1} h, \map {f_2} g *_2 \map {f_2} h}\) | Definition of Group Direct Product $\struct {H_1 \times H_2, *}$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \tuple {\map {f_1} {g \circ h}, \map {f_2} {g \circ h} }\) | $f_1$ and $f_2$ are Group Homomorphisms | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map {\paren {f_1 \times f_2} } {g \circ h}\) | Definition of Direct Product of $f_1$ and $f_2$ |

$\blacksquare$

## Sources

- 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{II}$: Groups: Direct Products