A homomorphism of groups is the first formal way to compare two groups without requiring their elements to look alike. The focus keyword homomorphism of groups means a function that carries products in the domain to products in the codomain. Students often check only where individual elements go, but the essential test always involves two arbitrary elements combined by the operation. Homomorphisms are later used to build kernels, quotient groups, isomorphism theorems, and permutation representations. In this lesson we define the condition, learn how to read it in different notations, and practise both proving and disproving that a function is a homomorphism of groups.

Let $(G,\ast)$ and $(G_1,\circ)$ be groups. A function $f:G\to G_1$ is called a \textbf{homomorphism of groups} if $$\begin{equation*} f(a\ast b)=f(a)\circ f(b) \end{equation*}$$ for all $a,b\in G$. When both operations are written multiplicatively, this condition is written as $$\begin{equation*} f(ab)=f(a)f(b). \end{equation*}$$