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:

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