Canonical Injection on Group Direct Product is Monomorphism

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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 1

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 $\inj_2$.

$\blacksquare$


Proof 2

A specific instance of Canonical Injection is Monomorphism, where the algebraic structures in question are groups.

$\blacksquare$


Sources