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.

  • $\begingroup$ I know that but why all automorphism of $S_3$ are inner only That was my question.Can you help me please? $\endgroup$ – MathLover May 2 '18 at 10:41
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    $\begingroup$ Any automorphism of $S_3$ must permute its three elements of order $2$. Since they generate $S_3$ this means that there are at most $6$ automorphisms. But there are $6$ inner automorphisms. $\endgroup$ – Derek Holt May 2 '18 at 11:05
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    $\begingroup$ See here for the case $n \ne 6$. $\endgroup$ – Derek Holt May 2 '18 at 11:07

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