Homomorphisms and Isomorphisms

A homomorphism between groups $(G, *)$ and $(H, \cdot)$ is a map $\varphi : G \to H$ that respects the operations:

It automatically sends identity to identity, $\varphi(e_G) = e_H$, and inverses to inverses. A bijective homomorphism is an isomorphism — the two groups are then structurally identical, just relabelled.

Worked example. The reduction map $\varphi : \mathbb{Z} \to \mathbb{Z}_5$, $\varphi(n) = n \bmod 5$, is a homomorphism from $(\mathbb{Z},+)$ to $(\mathbb{Z}_5,+)$: the remainder of a sum is the sum of the remainders. Its kernel — the elements sent to $0$ — is the set of multiples of $5$. Another classic: $\varphi(x) = 2x$ from $(\mathbb{R},+)$ to $(\mathbb{R},+)$, since $2(x+y) = 2x + 2y$.