Canonical Injection on Group Direct Product is Monomorphism/Proof 2
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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
A specific instance of Canonical Injection is Monomorphism, where the algebraic structures in question are groups.
$\blacksquare$