Solution
= Solution
The identity belongs to the <centralizer> $C_G(H)$. If $x,y\in C_G(H)$, then for every $h\in H$,
$$
(xy)h=x(yh)=x(hy)=(xh)y=h(xy).
$$
Also $xh=hx$ implies $h x^{-1}=x^{-1}h$ by multiplying on both sides by $x^{-1}$. Hence $xy$ and $x^{-1}$ belong to $C_G(H)$, proving by the <subgroup test> that $\boxed{C_G(H)\le G}$. This argument works even when $H$ is any subset, giving the <centralizer of a subset>.