# Group of Order less than 6 is Abelian

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## Contents

## 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