Does the conjugacy class of a coset consist of the cosets of elements in the conjugacy class?

Question

Say $H\triangleleft G$ and $xH\in G/H.$ Is the conjugacy class of $xH$ simply $\{yH:y\in [x]_G$}? Certainly these cosets are in the conjugacy class, but can there be others?

Answer 1

That is true because if $X=gH$, then $XyHX^{-1}=gHyHHg^{-1}=gyg^{-1}H$ since $H$ is a normal subgroup.

Answer 2

The elements of the conjugacy class of $xH$ are of the form $g^{-1}H \cdot xH \cdot gH=g^{-1}xgH$, since $H \unlhd G$. So yes it is simply what you are thinking.

However, if an element $g \in G$ centralizes $x$, then of course $\overline{g}$ will centralize $\overline{x}$ (where the overbar denotes modding out by $H$). But the converse is not true, $\overline{C_G(x)}$ can be proper in $C_{G/H}(\overline{x})$. This also implies that $|Cl_{G/H}(\overline{x})| \leq |Cl_G(x)|$.

On the other hand, it can also be proved that $|Cl_G(x)| \leq |H||Cl_{G/H}(\overline{x})|$.