Group of Order less than 6 is Abelian

Theorem
All groups with less than $6$ elements are abelian.

Proof
Let $G$ be a non-abelian group.

From Non-Abelian Group has Order Greater than 4, the order of $G$ must be at least $5$.

But $5$ is a prime number.

By Group of Prime Order is Cyclic it follows that a group of order $5$ is cyclic.

By Cyclic Group is Abelian this group is abelian.

Hence the result.

Also see

 * Symmetric group $S_n$ and dihedral group $D_n$: smallest non-abelian groups of order $6$ being $S_3$ and $D_3$.