Definition – Relation. – Math-reference.net

Contents
1. Definition
2. Types of homomorphism

Definition

Let $<G,*>$ and $<H,\times>$ be groups. Let $\phi :G\rightarrow H$ be a function. We say that $\phi$ is a homomorphism if

$\forall{g_1,g_2}\in{G},~~ \phi(g_1*g_2) = \phi(g_1)\times{\phi(g_2)}$

Alternatively, we can also define

Types of homomorphism

If the function $\phi :G \rightarrow{H}$ is bijective then we refer to $\phi$ as an isomorphism.