Homomorphism

\(\phi: G \to H\) preserving operation: \(\phi(ab) = \phi(a)\phi(b)\).

• Kernel: \(\ker\phi = \{g \mid \phi(g)=e\}\)
• Image: \(\operatorname{im}\phi = \phi(G)\)

Isomorphism Theorem

\(G/\ker\phi \cong \operatorname{im}\phi\)