I can show Inner Automorphism of $S_3$ are isomorphic to $S_3$ .Why every automorphism of $S_3$ is inner automorphism only? That is why there is no automorphism of $S_3$ exist that is not inner automorphism .
Why this is not in case of $S_6$?
Thanks in advanced.
2026-03-25 07:45:03.1774424703
Why every automorphism of $S_3$ is inner Automorphism But This is not case with $S_6$?
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In $S_3$, every automorphism must fix all the conjugacy classes, because distinct conjugacy classes correspond to distinct elements' orders (automorphisms preserve elements' order). So, each automorphism is class-preserving, hence cycle type-preserving. In particular, it sends transpositions to transpositions, and hence it is inner.
In $S_4$ and $S_5$, the classes of the transpositions and the double transpositions have different sizes. So, again, every automorphism is transpositions-preserving (automorphisms preserve classes' size), hence it is inner.
On the other hand, $S_6$ fulfils the necessary conditions for a non-inner automorphism to exist, namely the existence of conjugacy classes of elements of order $2$, which have also the same size: the classes of transpositions and triple transpositions both have size $15$. This doesn't prove yet that such non-inner automorphisms exist (swapping the two classes), but at least makes plausible their existence, here.