Kernel of Group Homomorphism is not Empty

Theorem

Let $G$ and $H$ be groups whose identity elements are $e_G$ and $e_H$ respectively.

Let $\phi: G \to H$ be a homomorphism from $G$ to $H$.

Let $\map \ker \phi$ denote the kernel of $\phi$.

Then:

$\map \ker \phi \ne \O$

where $\O$ denotes the empty set.

Proof

From Identity is in Kernel of Group Homomorphism we have that:

$e_G \in \map \ker \phi$

Hence the result.

$\blacksquare$

Proof

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 65$