Image of Canonical Injection is Normal Subgroup
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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:
- $\inj_1: \struct {G_1, \circ_1} \to \struct {G_1 \times G_2, \circ}$ 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.
$\blacksquare$
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Direct Products