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Let $H$ be a subgroup of finite index in group $G$ and let $g \in G$.

Question: Is it true, that $C_H(g)$ has finite index in $C_G(g)$ for all $g \in G$?

Here $C_H(g) := \{ h \in H\mid gh = hg\}$ denotes the centralizer of $g$ in $H$.

I know that the statement is true for $g \in H$. In this case it is a byproduct of the theory of ranks of projective modules over group rings. For, there one shows that for each conjugacy class $[h]$ of $H$ there is an integer $n_h$ such that for each projective $\mathbb{Z}G$-module $P$ the Hattori-Stallings rank satisfies:

$$R_H(P) = \sum_h n_h R_G(P)(h)\cdot [h]$$ where $h$ runs over a system of representatives of the conjugacy classes of $H$. Moreover, one can show $n_h = (C_G(h):C_H(h))$ (cf. Kenneth S. Brown: Cohomology of Groups, IX, Prop. 4.1 and Exercise 2).

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    $\begingroup$ Yes, this follows from the more general result that if $H$ is a subgroup of finite index in $G$ and $K$ is any subgroup of $G$ then $H\cap K$ has finite index in $K$. $\endgroup$ – Derek Holt Aug 7 '19 at 8:20
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    $\begingroup$ And note that $C_H(g)=H \cap C_G(g)$ $\endgroup$ – Nicky Hekster Aug 7 '19 at 8:25
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    $\begingroup$ @Derek Holt & Nicky Hekster: Great! Thank you very much. $\endgroup$ – tj_ Aug 7 '19 at 9:03
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    $\begingroup$ I find it hard to find a more sophisticated proof (using rank) of such an immediate fact! $\endgroup$ – YCor Aug 7 '19 at 13:47

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