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**ThePerfectHacker** Remember that $\displaystyle S_3 = \{ e,a,a^2,b,ab,a^2b\}$ where $\displaystyle a=(123),b=(12)$. Any automorphism on $\displaystyle S_3$ is determined by $\displaystyle \theta(a),\theta(b)$. The order of $\displaystyle \theta(a)$ is equal to order of $\displaystyle a$, thus, $\displaystyle \theta(a) = a,a^2$ similarly $\displaystyle \theta(b) = b,ab,a^2b$. In total we have at most $\displaystyle 6$ automorphism. Show that each one extens to an automorphism of $\displaystyle S_3$.