# Thread: Prove that I(G) is normal in A(G)

1. ## Prove that I(G) is normal in A(G)

For any group $G$ prove that $I(G)$ is a normal subgroup of $A(G)$.

Where $I(G)$ denotes the group of inner automorphisms and $A(G)$ denotes the group of automorphisms on a group $G$.

Thanks and Regards,
Kalyan.

2. Originally Posted by kalyanram
For any group $G$ prove that $I(G)$ is a normal subgroup of $A(G)$.

Where $I(G)$ denotes the group of inner automorphisms and $A(G)$ denotes the group of automorphisms on a group $G$.

Thanks and Regards,
Kalyan.

If $I_g\in I(G) (\,\,i.e.\,,\,I_g(x):=gxg^{-1}\,\,\forall x\in G)\,,\,\,then\,\,\forall \phi\in A(G)\,,\,\,\phi^{-1}I_g\phi=I_{\phi^{-1}(g)}$

Tonio