Image of Canonical Injection is Normal Subgroup

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:
 * $\inj_1: \struct {G_1, \circ_1} \to \struct {G_1 \times G_2, \circ}$ 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:
 * $(1): \quad \Img {\inj_1} \lhd \struct {G_1 \times G_2, \circ}$
 * $(2): \quad \Img {\inj_2} \lhd \struct {G_1 \times G_2, \circ}$

That is, the images of the canonical injections are normal subgroups of the group direct product of $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$.

Proof
From Image of Canonical Injection is Kernel of Projection:


 * $\Img {\inj_1} = \map \ker {\pr_2}$
 * $\Img {\inj_2} = \map \ker {\pr_1}$

That is:
 * the image of the (first) canonical injection is the kernel of the second projection
 * the image of the (second) canonical injection is the kernel of the first projection.

The domain of the projections is $G_1 \times G_2$, by definition.

The result follows from Kernel is Normal Subgroup of Domain.