Kernel

If $\phi :G\to H$ is a group homomorphism, the kernel is

$\ker (\phi )=\{g\in G:\phi (g)=e_H\}$.

The kernel measures how much information the map collapses.

A homomorphism is injective exactly when its kernel is trivial.

One-line intuition

The kernel is the part of the group that becomes invisible under the homomorphism.