Canonical Injection on Group Direct Product is Monomorphism/Proof 1
Theorem
Let $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$ be groups with identities $e_1, e_2$ respectively.
Let $\struct {G_1 \times G_2, \circ}$ be the group direct product of $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$
Then the canonical injections:
- $\inj_1: \struct {G_1, \circ_1} \to \struct {G_1, \circ_1} \times \struct {G_2, \circ_2}: \forall x \in G_1: \map {\inj_1} x = \tuple {x, e_2}$
- $\inj_2: \struct {G_2, \circ_2} \to \struct {G_1, \circ_1} \times \struct {G_2, \circ_2}: \forall x \in G_2: \map {\inj_2} x = \tuple {e_1, x}$
are group monomorphisms.
Proof
From Canonical Injection is Injection we have that the canonical injections are in fact injective.
It remains to prove the morphism property.
Let $x, y \in G_1$.
Then:
\(\ds \map {\inj_1} {x \circ_1 y}\) | \(=\) | \(\ds \tuple {x \circ_1 y, e_2}\) | Definition of $\inj_1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {x \circ_1 y, e_2 \circ_2 e_2}\) | Definition of Identity Element | |||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {x, e_2} \circ \tuple {y, e_2}\) | Definition of Group Direct Product | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\inj_1} x \circ \map {\inj_1} y\) | Definition of $\inj_1$ |
and the morphism property has been demonstrated to hold for $\inj_1$.
Thus $\inj_1: \struct {G_1, \circ_1} \to \struct {G_1, \circ_1} \times \struct {G_2, \circ_2}$ has been shown to be an injective group homomorphism and therefore a group monomorphism.
The same argument applies to $\inj_2$.
$\blacksquare$