Prove every group of order less or equal to five is abelian Is it possible to prove that every group of order less or equal to five is abelian?   
We know that groups of prime order are cyclic and therefore commutative. As the number $4$ is the only composite number $\le5$, it basically remains to show this for groups of order four.

$xy \neq yx \implies G$ has at least five elements: $e,x,y,xy,yx$. But $|G| = 5$ means that $G$ is isomorphic to $\mathbb{Z}_5$, so every group of order less than 6 must be abelian.
