Homomorphisms

Why

We name a function which preserves algebraic structure.

Definition

A group homomorphism between two groups $(A, +)$ and $(B, \tilde{+})$ is a bijection $f: A \to B$ such that $f(1_A) = 1_B$ for $1_A \in A$ and $1_B \in B$ and $f(a + a') = f(a) \tilde{+} f(a')$ for all $a, a' \in A$. We define a ring homomorphism and field homomorphism similarly.