If $G$ is a finite group s.t. $|G|=4$, is it abelian ? If $G$ is a finite group s.t. $|G|=4$, is it abelian ? To me it's isomorphic to $\mathbb Z/2\mathbb Z\times \mathbb Z/2\mathbb Z$ or $\mathbb Z/4\mathbb Z$, but a friend of me said that they are not the only group of order 4, and there exist some non-abelian. Do you have such example ?

If $G$ has  an element of order $4$ then $G=\mathbf{Z}/4$ otherwise $G$ has at least two elements of order $2$ $a,b$, $bab^{-1}=a$ implies that $G=\mathbf{Z}/2\times \mathbf{Z}/2$, $bab^{-1}=b$ implies $ab^{-1}=1$ thus $b=a$ impossible, $bab^{-1}=1$ implies $ba=b, a=1$ impossible.
