Group of Order less than 6 is Abelian
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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
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