# Why every automorphism of $S_3$ is inner Automorphism But This is not case with $S_6$?

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 $S_6$?
• I know that but why all automorphism of $S_3$ are inner only That was my question.Can you help me please? – MathLover May 2 '18 at 10:41
• 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. – Derek Holt May 2 '18 at 11:05
• See here for the case $n \ne 6$. – Derek Holt May 2 '18 at 11:07