Lemma: Kernel and Image of Group Homomorphism 

Let \((G,\ast)\) and \((H,\cdot)\) be two groups with the respective identities \(e_G\) and \(e_H\) and \(f:G\rightarrow H\) be a group homomorphism.

We define:

• the kernel $\operatorname{ker}(f):= \{g\in G\mid f(g)=e_H\}$, in other words, it is the fiber of $e_H$ under the group homomorphism $f$,
• the image $\operatorname{im}(f):=f[G]=\{f(g)\in H\mid g\in G\},$ in other words, it is the image of $G$ under the group homomorphism $f$.

The kernel and the image of $f$ fulfill the following defining properties:

1. \(\ker(f)=\{e_G\}\) \(\Longleftrightarrow f\) is injective.
2. \(\operatorname{im}(f)=H\) \(\Longleftrightarrow f\) is surjective.

| | | | | created: 2014-08-28 22:42:23 | modified: 2020-06-26 17:29:10 | by: bookofproofs | references: [696]

1.Proof: (related to "Kernel and Image of Group Homomorphism")