= Outer automorphism of a group
{title2=$\alpha\in\operatorname{Aut}(G)\setminus\operatorname{Inn}(G)$}
= Outer automorphism
{synonym}
An outer <group automorphism> is not <conjugation> by any element of the <group>. Its nonidentity class belongs to $\operatorname{Out}(G)=\operatorname{Aut}(G)/\operatorname{Inn}(G)$. A permutation-group example is the exceptional automorphism of $S_6$ that swaps its natural and transitive copies of $S_5$; it sends a <transposition> to a product of three disjoint <transpositions> and hence cannot be inner.
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