The conjugation Group action is continuous

Question

How can I prove that the group action from $G\times G\to G$ defined by $(g,x)\mapsto gxg^{-1}$ is a continuous function? I tried to use the known facts that multiplication and $(x,y)\mapsto xy^{-1}$ are continuous functions but without success.

Thank you

Answer 1

Since the map $$G\times G\to G\\(x,y)\mapsto xy^{-1}$$ is continuous, the following compositions are as well

\begin{matrix} G\times G&\to& G\times (G\times G)\times G&\to& (G\times G)\times G&\to& G\times G&\to& G\\(g,x)&\mapsto&(g,(e,x),g)&\mapsto&((g,x^{-1}),g)&\mapsto& (gx,g)&\mapsto&gxg^{-1}\end{matrix}