Question

Let G be a group, and let g ϵ G. Let φg : G → G be defined by φg(x) = gxg-1 for x ϵ G. For which g ϵ G is φg a homomorphism?

Answer 1

It turns out \(\phi_{g}\) is a homomorphism for all \(g\in G\). You need to verify that: $$\phi _{g}(xy)=\phi_{g}(x)\phi_{g}(y)$$for all \(x,y\in G\), which isn't too hard to do using the definition of \(\phi_{g}\) and the fact that \(g^{-1}g=e\) for all \(g\in G\).