Please help: How to show that Aut(G) is a subgroup of Sym(G)

Question

I have to prove that if G is a group, Aut(G) is a subgroup of Sym(G), and later that the inner automorphisms group is a normal subgroup of Aut(G). I have time until tomorrow in the morning, but I have no idea what to do, please please please help me.

Answer 1

Here is a roadmap:

• $\operatorname{Aut}(G) \subseteq \operatorname{Sym}(G)$

• $id \in \operatorname{Aut}(G)$

• $\phi \in \operatorname{Aut}(G) \implies \phi^{-1} \in \operatorname{Aut}(G)$

$$\phi^{-1}(xy)=\phi^{-1}(\phi(x')\phi(y'))=\phi^{-1}(\phi(x'y'))=x'y'=\phi^{-1}(x)\phi^{-1}(y)$$

• $\phi, \psi \in \operatorname{Aut}(G) \implies \phi\circ\psi \in \operatorname{Aut}(G)$

Which part are you having trouble with?