# Finite index of centralizer $C_H(g)$ in $C_G(g)$

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).

• 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 $K$. – Derek Holt Aug 7 '19 at 8:20
• And note that $C_H(g)=H \cap C_G(g)$ – Nicky Hekster Aug 7 '19 at 8:25
• @Derek Holt & Nicky Hekster: Great! Thank you very much. – tj_ Aug 7 '19 at 9:03
• I find it hard to find a more sophisticated proof (using rank) of such an immediate fact! – YCor Aug 7 '19 at 13:47