Consider the action of a group $G$ of order $900$ on a set $X$ with cardinality $15$. By using the Orbit-Stabiliser Theorem, what is the value for the order of the stabiliser of a point of $X$?
So I know that the Orbit-Stabiliser Theorem states that: $$ |G|=|\operatorname{Stab}(x)|\times |\operatorname{Orb}(x)| $$ But I'm not sure this can be applied to the problem at hand, nor how to approach this question. Any help would be great thanks!
Stabilizers are subgroups, so $80$ and $120$ are ruled out by Lagrange's theorem. Then, $45$ would lead to $|O(x)|=20$, which can't be because the orbits partiton $X$, whose size is $15$. So you are left with $150$ as possible order of a stabilizer.