$\newcommand{\stab}{\text{stab}}$ Proposition: Let $G$ be a group acting on a finite set $X$. Let $N$ be the number or orbits of $X$ under this action, and for each orbit let $x_i \in X$ be some element in the orbit. Then $$ |X| = \sum_{i=1}^N[G:\stab(x_i)]. $$
The proof is a direct appication of the orbit-stabilizer theorem but why do we need to require that $|X|< \infty$ here? If $X$ was an infinite then at least one of the orbits is going to be infinite because the orbits partition $X$, but then won't this still hold? Sorry I assume I'm missing something evident. Thanks in advance for the clarification.