Identity is in Kernel of Group Homomorphism
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Theorem
Let $G$ and $H$ be groups.
Let $e_G$ and $e_H$ be the identity elements of $G$ and $H$ respectively.
Let $\phi: G \to H$ be a (group) homomorphism from $G$ to $H$.
Then:
- $e_G \in \map \ker \phi$
where $\map \ker \phi$ is the kernel of $\phi$.
Proof
From the definition of kernel:
- $\map \ker \phi = \set {x \in G: \map \phi x = e_H}$
From Group Homomorphism Preserves Identity we have that:
- $\map \phi {e_G} = e_H$
Hence the result.
$\blacksquare$
Proof
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 65$