Constant Mapping to Identity is Homomorphism/Groups

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Let $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$ be groups with identities $e_1$ and $e_2$ respectively.

Let $\phi_e: \struct {G_1, \circ_1} \to \struct {G_2, \circ_2}$ be the constant mapping defined as:

$\forall x \in G_1: \map {\phi_e} x = e_2$

Then $\phi_e$ is a group homomorphism whose image is $\set {e_2}$ and whose kernel is $G_1$.


Let $x, y \in G_1$.


\(\ds \map {\phi_e} {x \circ_1 y}\) \(=\) \(\ds e_2\) as $x \circ_1 y \in G_1$
\(\ds \) \(=\) \(\ds \map {\phi_e} x \circ_2 \map {\phi_e} y\) as $\map {\phi_e} x = e_2$ and $\map {\phi_e} y = e_2$

So $\phi_e$ is a group homomorphism.

The results about image and kernel follow directly by definition.