Canonical Injection is Right Inverse of Projection
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Theorem
Let $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$ be groups with identity elements $e_1$ and $e_2$ respectively.
Let $\struct {G_1 \times G_2, \circ}$ be the group direct product of $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$
Let:
- $\pr_1: \struct {G_1 \times G_2, \circ} \to \struct {G_1, \circ_1}$ be the first projection from $\struct {G_1 \times G_2, \circ}$ to $\struct {G_1, \circ_1}$
- $\pr_2: \struct {G_1 \times G_2, \circ} \to \struct {G_2, \circ_2}$ be the second projection from $\struct {G_1 \times G_2, \circ}$ to $\struct {G_2, \circ_2}$.
Let:
- $\inj_1: \struct {G_1, \circ_1} \to \struct {G_1 \times G_2, \circ}$ be the canonical injection from $\struct {G_1, \circ_1}$ to $\struct {G_1 \times G_2, \circ}$
- $\inj_2: \struct {G_2, \circ_2} \to \struct {G_1 \times G_2, \circ}$ be the canonical injection from $\struct {G_2, \circ_2}$ to $\struct {G_1 \times G_2, \circ}$.
Then:
- $(1): \quad \pr_1 \circ \inj_1 = I_{G_1}$
- $(2): \quad \pr_2 \circ \inj_2 = I_{G_2}$
where $I_{G_1}$ and $I_{G_2}$ are the identity mappings on $G_1$ and $G_2$ respectively.
Proof
This is a specific instance of External Direct Product of Projection with Canonical Injection, where the algebraic structures in question are groups.
$\blacksquare$
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Direct Products