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Let $G$ be a finite group and $N\lhd G$. For every $x\in G-N$ we have $C_G(x)N/N\leq C_{G/N}(Nx)$ where $C_G(x)$ is the centralizer of $x$ in $G$. But I don't know what is the qutient.

In fact I want to know the $[C_{G/N}(Nx):C_G(x)N/N]$. It is more important when $N=Z(G)$.

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    Well the answer depends on $G$, $N$ and $x$. It is not clear what kind of answer your are looking for!2012-12-23

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