Alternating Group on More than 3 Letters is not Abelian

Theorem
Let $n$ be an integer such that $n > 3$.

Then the $n$th alternating group $A_n$ is not abelian.

Proof
Let $\tuple {1, 2, 3}$ and $\tuple {2, 3, 4}$ be elements of $A_n$.

Then we have:

So:
 * $\tuple {1, 2, 3} \tuple {2, 3, 4} \ne \tuple {2, 3, 4} \tuple {1, 2, 3}$

and so $A_n$ is not abelian.