The motivation of this self made problem is to get a better understanding of Group actions.
Say $G$ is a group that acts on a set $X$ of 4 elements such that the action of $G$ has 2 orbits.
What I want to do is work back and figure out what is the structure of $G$ from the above statement. Also, setup explicit actions ( if possible ).
My results
From definition, $G$ is isomorphic to a subgroup of $S_4$. As there are 2 orbits under the action of $G$ and orbits partition $X$, we can infer that the orbits are of the form i) $\{a_1,a_2\},\{a_3,a_4\}$ or ii) $\{a_1,a_2,a_3\},\{a_4\}$. For both cases let $a_1$ and $a_4$ be representatives of the respective orbits.
If the orbits are of the form (ii), then by the Orbit-Stabilizer theorem we have that $|\text{Stab}_{G}(a_4)| = |G|$ and $|\text{Stab}_{G}(a_1)| = |\text{Stab}_G(a_4)|/3$. By Cauchy's theorem, we have that $|Stab_G(a_4)|$ has an element of order 3. Hence, it has a cyclic subgroup $<g>$ of order 3.
If the orbits are of the form (i), then by the the Orbit-Stabilizer theorem we have that $|\text{Stab}_{G}(a_4)| = |\text{Stab}_{G}(a_1)|$.
Now I am stuck. I am at a loss, what extra information can I deduce that will allow me to achieve my goal ?
By explicitly taking subgroups of $S_4$, and applying them on $X$, I have that $S_3$, $<(123)>$ and $<(12)(34)>$ satisfy the above condition ( there may be more but I have not computed all of them ).
Note
- I would like to avoid applying Sylow's theorems or Burnside's theorem at this point of time. The class equation is okay.