Canonical Injection on Group Direct Product is Monomorphism/Proof 1

Theorem
Let $\left({G_1, \circ_1}\right)$ and $\left({G_2, \circ_2}\right)$ be groups with identities $e_1, e_2$ respectively.

Let $\left({G_1 \times G_2, \circ}\right)$ be the group direct product of $\left({G_1, \circ_1}\right)$ and $\left({G_2, \circ_2}\right)$

Then the canonical injections:


 * $\operatorname{in}_1: \left({G_1, \circ_1}\right) \to \left({G_1, \circ_1}\right) \times \left({G_2, \circ_2}\right): \forall x \in G_1: \operatorname{in}_1 \left({x}\right) = \left({x, e_2}\right)$


 * $\operatorname{in}_2: \left({G_2, \circ_2}\right) \to \left({G_1, \circ_1}\right) \times \left({G_2, \circ_2}\right): \forall x \in G_2: \operatorname{in}_2 \left({x}\right) = \left({e_1, x}\right)$

are group monomorphisms.

Proof
From Canonical Injections are Injections we have that the canonical injections are in fact injective.

It remains to prove the morphism property.

Let $x, y \in \left({G_1, \circ_1}\right)$.

Then:

and the morphism property has been demonstrated to hold for $\operatorname{in}_1$.

Thus $\displaystyle \operatorname{in}_1: \left({G_1, \circ_1}\right) \to \left({G_1, \circ_1}\right) \times \left({G_2, \circ_2}\right)$ has been shown to be an injective group homomorphism and therefore a group monomorphism.

The same argument applies to $\operatorname{in}_2$.