Finding an example for a group operation on a set

Question

An operation of a group $G$ on a set $S$ is a rule $G \times S \to S$. Suppose we have $G=C_3 \times C_5 = C_{15}$, and $S$ is a set with $4$ elements. I think we can partition $S$ into orbits $O_1$ and $O_2$ with $|O_1| = 1$ and $|O_2| = 3$. By the counting formula, the stabilizer of $O_1$ has order $15$ and the stabilizer of $O_2$ has order $5$. Is that reasoning correct? How can we find an example where this is the case?

Answer 1

Sounds like the concept you are looking for is group action on a set.

If the set is finite, all group action on it amounts to permuting the elements of the set. So, all that is needed is to associate, homomorphically, to each element of $C_{15}$ a permutation of $$ S = \{s_{1}, s_{2}, s_{3}, s_{4}\}. $$ So, if $c$ is a generator of the cyclic group $C_{15}$, let's assign to it the permutation $$ (s_{1}) \quad ( s_{2} \; s_{3} \; s_{4}). $$ It fixes $s_{1}$ and cyclically permutes the other elements of $S$. Now you can compute the permutations corresponding to $c^2, c^3, $ etc..