I need to prove the following fact: if $G$ is a non abelian group of order $6$, then $G$ is isomorphic to the dihedral group $D_3$. Here what I have done:
Suppose that $G$ is non abelian group of order $6$. Since $G$ is not abelian, it cannot have a element of order $6$. (This is correct, since if $G$ had a element of order $6$, then $G$ would be cyclic, and hence abelian, a contradiction). Also, it cannot have elements of order 1, by the same argument as before.
But I got stucked in here. I'm struggling with the two other possibilities of the order of the elements in $G$: order $2$ and $3$. Any help or hints that you can bring me would be appreciated.