I need to find all of the isomorphism classes of transitive actions for $\mathbb{Z}_{4}$. I know that they are in a bijection with the conjugacy classes, and that since the group is abelian then every element is in its own conjugacy class, and therefore there are four isomorphism classes. However, I'm not sure how to actually write down the isomorphism class, or any representative of the class. I imagine that I get to choose whatever $G$-set I want, so that I could create the following representative of one isomorphism class:
With the $G$-set $\{0\}$, take the action $g(0)=0$ for all $g\in G$.
Then if we take a two-element $G$-set like $\{0,1\}$ then we could take the action $0(n) = n=2(n)$ and $1(n)=3(n) = n+1\mod{2}$.
Is this all basically the right idea? Do I just continue in this direction?
Lemma:Every transitive action correspond to the action of $G$ on left cosets of $H$ with left multiplication where $H\leq G$.
Since $Z_4$ has three subgroups $e,Z_2,Z_4$ we have three different isomorphism classs.
İf you wonder the proof of the lemma, it is a corolory of orbit stabilizer theorem. The trick is that there is a one to one correspondance between left cosets of $Stab(\alpha)$ and
elments of acted set $\Omega$.