Prove that the symmetric group of degree at least 3 is non-abelian

The symmetric group $S_n$ is non-abelian for $n \geq 3$ .

Proof. Since we need to look at degrees at least

3

, we can take

(12),(13) \in S_n

. Now note that:

\begin{align*} (12)(13) = (132) \end{align*}

and

\begin{align*} (13)(12) = (123). \end{align*}

In order to be abelian, it must be that

(12)(13) = (13)(12)

, but that is not the case since

\begin{align*} (12)(13) = (132) \neq (123) = (13)(12). \end{align*}

Therefore, the symmetric group

S_n

is non-abelian for

n \geq 3