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

By Cyclic Group is Abelian this group is abelian.

Hence the result.

$\blacksquare$

Also see

• Symmetric Group on 3 Letters
• Definition:Symmetry Group of Equilateral Triangle

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 29 \beta$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 44.3$ Some consequences of Lagrange's Theorem