Conjugation

Conjugation on a group $G$ by an element $x \in G$ is the homomorphism defined by

$\phi\ x\ g = x\ g\ x^{-1}.$

$\phi\ x$ is clearly a homomorphism:

$\phi\ x\ (g\ h) = x\ g\ h\ x^{-1} = x\ g\ x^{-1}\ x\ h\ x^{-1} = (\phi\ x\ g)\ (\phi\ x\ h).$