$|$Inn$(S_2)|$ and $|$Aut$(S_2)|$

Question

Is it true that $|$Inn$(S_2)|=1$? This is what I calculated but I want to double-check.

Also, does $|$Aut$(S_2)|=1$? I can't think of a non-inner automorphism of $S_2$.

Answer 1

As said DonAntonio in the comment above, $S_2$ is just the cyclic group of order $2$, isomorphic to $(\{-1,+1\}, \cdot)$.

An automorphism $f : S_2 \to S_2$ must map $(1)$ to $(1)$, and since $f$ is injective, it must map $(1 \; 2)$ to $(1 \; 2)$. Therefore $f$ is the identity of $S_2$.

Any inner automorphism of $S_2$ is an automorphism of $S_2$, so it is the identity. This holds also because $S_2$ is abelian, so any inner automorphism of $S_2$ is the identity of $S_2$.