TITLE: Prove every group of order less or equal to five is abelian
QUESTION [8 upvotes]: 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.

REPLY [4 votes]: $xy \neq yx \implies G$ 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.