Cayley's Theorem: Every group is isomorphic to a group of permutations.
$\mathbb Z_6$ is a group and $S_3$ is a permutation, but $\mathbb Z_6$ is not isomorphic to $S_3$.
$\mathbb Z_6$ is abelian while $S_3$ is not, hence they are not isomorphic.
So, what is going on here? I thought every group is isomorphic to a group of permutations.
Best Answer
A more precise statement of Cayley's theorem states that if $|G| = n$, then $G$ is a subgroup of $S_n$.
In this case, $|\mathbb{Z}_6| = 6$, so $\mathbb{Z}_6 \leq S_6$. In particular, it will be the subgroup generated by the $6$-cycle $\sigma = (1,2,3,4,5,6)$.