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What is an example of an automorphism of a group G that does not belong to Inn(G), the group of all inner automorphisms?

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Any nontrivial automorphism of an abelian group, for starters. – Henning Makholm Mar 1 '14 at 1:24
Could you give me an example? – Paul Malinowski Mar 1 '14 at 1:24
up vote 7 down vote accepted

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 $(1\,2)\notin A_4$).

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I would say that your second example non-inner because $C_{S_4}(A_4) = \{1\}$ and $(12) \not\in A_4$. For example $(56) \not\in A_4$, but in $A_4$ conjugation by $(56)$ is the trivial automorphism, which is inner. – Mikko Korhonen Mar 1 '14 at 8:13

All inner automorphism of Abelian groups is identity. $f: g \rightarrow g^{-1}$ is a automorphism for Abelian groups.

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Perhaps it should be added that $f$ is trivial (and thus inner) if and only if the group is an elementary abelian $2$-group. – Andreas Caranti Mar 1 '14 at 11:14
Yes, you are right. – Wei Zhou Mar 1 '14 at 12:58

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.

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