Automorphism and Inner automorphism What is an example of an automorphism of a group G that does not belong to Inn(G), the group of all inner automorphisms?
 A: Pick any automorphism of an abelian group that is not the identity. For example, any invertible linear transformation $\mathbb R^n\to\mathbb R^n$ (which is not the identity) is a non-inner automorphism of $(\mathbb R^n,{+})$. Or, if you want a completely concrete example, how about $f(x)=x^3$ as an automorphism of $\mathbb R^\times$?
For non-abelian groups, the simplest example is probably the automorphism of $A_4$ given by conjugation by the transposition $(1\,2)$. (This is non-inner because $(1\,2)\notin A_4$).
A: All inner automorphism of Abelian groups is identity. $f: g \rightarrow g^{-1}$ is a automorphism for Abelian groups.
A: Let $F=\langle  x, y \rangle $ be a free group of rank 2 with $x$ and $y$ as generator words.
Then $\phi: F \rightarrow F$ with $\phi(x)=y$ and $\phi(y)=x$ induce an automorphism of $F$ which is not inner. (note that $F$ is free group so $x$ and $y$ can not be conjugate elements in $F$ since there is no relation in the presentation of a free group).
In fact this example can be extended to any finite-rank free group.
