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Problem Text
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 (Optional)
$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.
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