Problem source

Let $S_{3}$ be the group of permutations of three objects and $Z_{6}$
be the group of integers under addition modulo 6. List all the elements
of each group, stating the order of each element. State, with reasons,
whether $S_{3}$ is isomorphic with $Z_{6}.$

Let $C_{6}$ be the group of 6th roots of unity. That is, $C_{6}=\{1,\alpha,\alpha^{2},\alpha^{3},\alpha^{4},\alpha^{5}\}$
where $\alpha=\mathrm{e}^{\mathrm{i}\pi/3}$ and the group operation
is complex multiplication. Prove that $C_{6}$ is isomorphic with
$Z_{6}.$ Is there any (multiplicative or additive) subgroup of the
complex numbers which is isomorphic with $S_{3}$? Give a reason for
your answer.

Solution source

$S_3 $
$\begin{array}{c | c |c |c |c |c |c |}
\text{elements} & e & (12) & (13) & (23) & (123) & (132) \\
\text{order} & 1 & 2 & 2 & 2 & 3 & 3 \\
\end{array}$

$\mathbb{Z}_6$
$\begin{array}{c | c |c |c |c |c |c |}
\text{elements} & 0 & 1 & 2 & 3 & 4 & 5 \\
\text{order} & 1 & 6 & 3 & 2 & 3 & 6 \\
\end{array}$

$S_3$ is not isomorphic to $\mathbb{Z}_6$ since $\mathbb{Z}_6$ has two elements of order $6$ but $S_3$ has none.

Consider the map $f : \mathbb{Z}_6 \to C_6$ with $i \mapsto \alpha^i$. This is an isomorphism, since $i + j \mapsto \alpha^{i+j} = \alpha^i\alpha^j$

$S_3$ is non-abelian, since $(12)(123) = (23) \neq (13) = (123)(12)$ but multiplication and addition of complex numbers is commutative.