Statement#

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

Prove the First Isomorphism Theorem: $G/\ker(\phi) \cong \text{Im}(\phi)$

where $\ker(\phi) = \{g \in G : \phi(g) = e_H\}$ is the kernel and $\text{Im}(\phi) = \{\phi(g) : g \in G\}$ is the image.

Required Topics#

• Group homomorphisms
• Kernels and images
• Normal subgroups
• Quotient groups
• Isomorphisms

What to Prove#

1. $\ker(\phi)$ is a normal subgroup of $G$
2. Define an isomorphism $\bar{\phi}: G/\ker(\phi) \to \text{Im}(\phi)$
3. Show $\bar{\phi}$ is well-defined
4. Prove $\bar{\phi}$ is a homomorphism
5. Prove $\bar{\phi}$ is bijective (injective and surjective)

Solution#

Solution coming soon.