In the proof of Lemma 2.1 in


*Ershov, Mikhail; He, Sue*, [**On finiteness properties of the Johnson filtrations**](https://doi.org/10.1215/00127094-2018-0005), [ZBL06904638](https://zbmath.org/?q=an:06904638),


the authors claim the following (without proof).


*Let $G$ be a finitely generated group, and let $g\_1,\ldots,g\_k\in G$ be pairwise commuting elements, such that the union of the centralizers $S:=\bigcup\_{i=1}^k\mathrm{Cent}\_G(g\_i)$ generates $G$. Then it is possible to find a finite generating set for $G$ contained in $S$.*


I can't see why this should be true in general, but I don't have any counter-example either. Can anybody help me?



A group-theoretic lemma in a paper by Ershov and He
Here is @AndyPutman's [comment](https://mathoverflow.net/questions/435523/a-group-theoretic-lemma-in-a-paper-by-ershov-and-he#comment1122032_435523) as an answer (so that it can be accepted), made CW to avoid reputation. If @AndyPutman prefers to post the answer, then I will delete this.



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> This only needs the fact that $G$ is finitely generated and $S$ is a generating set. If $T$ is a finite generating set for $G$, then since $S$ is another generating set each $t$ in $T$ can be written as a word in $S$. Let $S'$ contain the letters that appear in such words as $t$ ranges over $T$. Then $S'$ is a finite generating set.
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