Let $H,K$ be subgroups of a group $G$. If $[H,K]$ is finite and $H$ is finitely generated, then the centralizer $C_K(H)$ has finite index in $K$. More precisely, if $[H,K]$ has order $n$ and $H$ is $m$-generated, then $|K : C_K(H)| \leq n^m$.
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