Relationship between $Aut(G)$ and Symmetric group on $G$ I have read that $Aut(G)$ is a subset of $S_g$.
So say I have a group $G = \{1, 2, 3\}$ for example. Then $S_G = S_3$ is the group of all permutation of the three elements of $G$.
But I don't see why $Aut(G)$ is a subset of $S_G$ as opposed to $Aut(G) = S_G$.
Each element of $S_3$ maps each element of $G$ to an element of $G$. I.e. each element is an automorphism. So why is $Aut(G) \subset S_3$ instead of $Aut(G) = S_3$?

There are elements in $\,\operatorname{Sym}_G\,$ which are not automorphisms of the group $\,G\,$, say the permutation $\,(01)\,$ in $\,S_3\,$ is not an automorphism of cyclic group $\,\Bbb Z_3:=\Bbb Z/3\Bbb Z:=\{0,1,2\}\,$, with operation modulo $\,3\,$
