= Solution
{c}
A left <group action> of $G$ on a <set> $S$ is a <function> $G\times S\to S$, written $(g,s)\mapsto g\cdot s$, satisfying
$$
\boxed{e\cdot s=s,\qquad(gh)\cdot s=g\cdot(h\cdot s)}
$$
for all $g,h\in G$, $s\in S$. The <orbit of a group action> through $s$ and the <stabilizer subgroup> of $s$ are, respectively,
$$
\boxed{G\cdot s=\{g\cdot s:g\in G\},\qquad G_s=\{g\in G:g\cdot s=s\}.}
$$
The <identity element> belongs to $G_s$. If $g,h\in G_s$, then $(gh)\cdot s=g\cdot(h\cdot s)=g\cdot s=s$, so $gh\in G_s$. If $g\in G_s$, applying $g^{-1}$ to $g\cdot s=s$ gives $s=g^{-1}\cdot s$, so $g^{-1}\in G_s$. These facts prove that the <stabilizer subgroup> is a <subgroup> of $G$.
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