= Solution
For a <group action>, the <orbit-stabilizer theorem> identifies the <orbit of a group action> of $x$ with the left <cosets> of its <stabilizer subgroup>:
$$
G/G_x\longrightarrow Gx,\qquad gG_x\longmapsto gx.
$$
This is well defined and bijective because $gx=hx$ exactly when $h^{-1}g\in G_x$. In particular, for finite $G$,
$$
\boxed{|Gx|=[G:G_x],\qquad |G|=|Gx||G_x|.}
$$
Under the <conjugation action> $g\cdot x=gxg^{-1}$, a singleton <orbit of a group action> means that $gxg^{-1}=x$ for every $g$. This is equivalent to $gx=xg$ for every $g$, so precisely these elements constitute the <centre of a group>
$$
Z(G)=\{x\in G:xg=gx\text{ for all }g\in G\}.
$$
The identity is central, and if $x,y$ commute with every element then so do $xy$ and $x^{-1}$. Thus $Z(G)$ is a <subgroup>. Each central element is fixed under every conjugation, giving $gZ(G)g^{-1}=Z(G)$: \b[the centre is a <normal subgroup>].
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