Inner automorphism/Automorphism/Fact/Proof

Proof

We have

${}\kappa _{g}(xy)=gxyg^{-1}=gxg^{-1}gyg^{-1}=\kappa _{g}(x)\kappa _{g}(y)\,,$

so that ${}\kappa _{g}$ this is a group homomorphism. We have

${}\kappa _{g}(\kappa _{h}(x))=\kappa _{g}(hxh^{-1})=ghxh^{-1}g^{-1}=ghx(gh)^{-1}=\kappa _{gh}\,.$

This implies, on one hand, that

${}\kappa _{g^{-1}}\circ \kappa _{g}=\kappa _{g^{-1}g}=\operatorname {Id} _{G}\,;$

therefore, ${}\kappa _{g}$ is bijective and an automorphism. On the other hand, this implies that the total mapping ${}\kappa$ is a group homomorphism.

To fact