A [[Group]] [[Homomorphism]] (an instance of [[Homomorphism]] for [[Group|Groups]]) is some [[Function]] $f$ with a [[Domain|domain]] of some group $(G,\circ)$ and [[Codomain|codomain]] of some group $(H, *)$, such that for some two elements $u,v\in G$, the [[Function|function]] $f$ maps $u \circ v$ to $f(u)*f(v)$ $\huge \begin{align} f&: G \to H \\ f(u\circ v) &= f(u) * f(v) \end{align} $ >[!example] >The [[Logorithm]] function $\ln$ is a [[Group Homomorphism]] between $(\R,+)$ to $(\R, \cdot)$. >$\huge \ln(x +y) = \ln(x) \cdot \ln(y) $