Let $N$ be a normal subgroup of $G$, and $|x^{G}|$ denote the size of the conjugacy class containing $x$. Please show the following statements:
If $x\in N$, then $\left \vert x^{N} \right \vert$ divides $\left \vert x^{G} \right \vert$
If $x\in G$, then $\left \vert ( xN )^{G/N} \right \vert$ divides $\left \vert x^{G} \right \vert$