Solution
= Solution
If $gH$ is a fixed coset, then
$$
xgH=gH.
$$
Multiplying on the left by $g^{-1}$ gives
$$
g^{-1}xgH=H,
$$
so $g^{-1}xg\in H$. Thus
$$
\boxed{x\text{ is conjugate in }G\text{ to an element of }H}.
$$
= Solution
If $gH$ is a fixed coset, then
$$
xgH=gH.
$$
Multiplying on the left by $g^{-1}$ gives
$$
g^{-1}xgH=H,
$$
so $g^{-1}xg\in H$. Thus
$$
\boxed{x\text{ is conjugate in }G\text{ to an element of }H}.
$$