I was wondering how to prove that ($\mathbb{Z}_{3}$, +), the group of integers under addition is isomorphic to $C_{3}$, the third cyclic group such that $\mathbb{Z}_{3} \cong C_{3}$.
Also that $\mathbb{Z}_{3}$ is a proper subgroup of $S_{3}$, the third group of permutations.
I understand that one can express ${C}_{3} = \{e, a, a^{2}\}$ where $a = \exp(2\pi i /3)$ and $a^{3} = e$ and $\mathbb{Z}_{3} = \{e, 1, 2\}$ but I am unsure how to go about explicitly expressing the isomorphism.
It must be possible to define a homomorphism such that $\phi: \mathbb{Z}_{3} \longmapsto C_{3}$, where
$\phi(g_{1}\cdot g_{2}) = \phi(g_{1})\star\phi(g_{2})$, $g_{1}, g_{2} \in \mathbb{Z}_{3}$.
Just consider the map $f\colon\mathbb{Z}_3\longrightarrow C_3$ such that $f(0)=1$, that $f(1)=a$, and that $f(2)=a^2$. In other words, $f(n)=a^n$.
And a natural injective map from $\mathbb{Z}_3$ into $S_3$ is $n\mapsto(1\ \ 2\ \ 3)^n$.