Image of Canonical Injection is Normal Subgroup

Theorem
Let $\left({G_1, \circ_1}\right)$ and $\left({G_2, \circ_2}\right)$ be groups with identity elements $e_1$ and $e_2$ respectively.

Let $\left({G_1 \times G_2, \circ}\right)$ be the group direct product of $\left({G_1, \circ_1}\right)$ and $\left({G_2, \circ_2}\right)$.

Let:
 * $\operatorname{in}_1: \left({G_1, \circ_1}\right) \to \left({G_1 \times G_2, \circ}\right)$ be the canonical injection from $\left({G_1, \circ_1}\right)$ to $\left({G_1 \times G_2, \circ}\right)$


 * $\operatorname{in}_2: \left({G_2, \circ_2}\right) \to \left({G_1 \times G_2, \circ}\right)$ be the canonical injection from $\left({G_2, \circ_2}\right)$ to $\left({G_1 \times G_2, \circ}\right)$.

Then:
 * $(1): \quad \operatorname{Im} \left({\operatorname{in}_1}\right) \lhd \left({G_1 \times G_2, \circ}\right)$
 * $(2): \quad \operatorname{Im} \left({\operatorname{in}_2}\right) \lhd \left({G_1 \times G_2, \circ}\right)$

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

Proof
From Image of Canonical Injection is Kernel of Projection:


 * $\operatorname{Im} \left({\operatorname{in}_1}\right) = \ker \left({\operatorname{pr}_2}\right)$
 * $\operatorname{Im} \left({\operatorname{in}_2}\right) = \ker \left({\operatorname{pr}_1}\right)$

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.