Past exam of the options II for mathematics and philosophy course of the University of Oxford / 2013 / B2 / a / Solution

The orbit of $s$ is $O_s=\{gs:g\in G\}$, and its stabilizer is $G_s=\{g\in G:gs=s\}$. The identity fixes $s$; if $g,h$ fix it then $(gh)s=g(hs)=s$; and $gs=s$ implies $g^{-1}s=s$. Thus $G_s$ is a subgroup.

A group action is transitive when it has just one orbit: for every $s,t\in S$ there is $g\in G$ with $gs=t$.