Solution
= 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 \b[$G_s$ is a <subgroup>].
A <group action> is <transitive group action>[transitive] when it has just one <orbit>: for every $s,t\in S$ there is $g\in G$ with $gs=t$.