Homomorphism

A homomorphism is a map $\varphi: G \to H$ between groups such that

$\varphi(ab) = \varphi(a)\varphi(b)$

for all $a,b \in G$.

It preserves the group structure: identity, inverses, and products.

One-line intuition

A homomorphism is a map that respects the multiplication law.