Past exam of the mathematics course of the University of Cambridge / 2011 / ia / Paper 3 / 7D / a / 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:This is well defined and bijective because $gx=hx$ exactly when $h^{-1}g\in G_x$. In particular, for finite $G$,Under the conjugation action $g\cdot x=gx{g}^{-1}$, a singleton orbit of a group action means that $gx{g}^{-1}=x$ for every $g$. This is equivalent to $gx=xg$ for every $g$, so precisely these elements constitute the centre of a groupThe 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 $gZ(G)g^{-1}=Z(G)$: the centre is a normal subgroup.