C6 is not Isomorphic to S3

Theorem

Let $C_6$ denote the cyclic group of order $6$.

Let $S_3$ denote the symmetric group on $3$ letters.

Then $C_6$ and $S_3$ are not isomorphic.

Proof

Note that both $C_6$ and $S_3$ are of order $6$.

From Cyclic Group is Abelian, $C_6$ is abelian.

From Symmetric Group is not Abelian, $S_6$ is not abelian.

From Isomorphism of Abelian Groups, if two groups are isomorphic, they are either both abelian or both not abelian.

Hence $C_6$ and $S_3$ are not isomorphic.

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 7.3$. Isomorphism: Example $136$