Let V denote the Klein 4-group. Show that $\text{Aut} (V)$ is isomorphic to $S_3$ [duplicate]

Solution 1:

Any automorphism $\phi$ of $V$ sends its identity element $e$ to itself. What is interesting is the way that $\phi$ rearranges the other three elements. The claim here is that any of the six possible bijections from $\tau, \tau_1, \tau_2$ to itself is in fact a group automorphism of $V$. This will give an isomorphism from $\operatorname{Aut}(V) \cong S_3$ that sends an automorphism $\phi \in \operatorname{Aut}(V)$ to its corresponding permutation of the three elements $\tau, \tau_1, \tau_2$.

Solution 2:

There are three elements of order $2$ in $V$. Identity element must go to identity element by any automorphism, and you can permute in any way the elements of order $2$, so $Aut(V)=S_3$.

Of course, you need to check that any permutation gives you an automorphism. It is obviously bijective, and is very easy to check directly that any such permutation preserves the group law.