I'm studying the paper "On the verbal width of finitely generated pro-p groups" by Andrei Jaikin-Zapirain ([link at ProjectEuclid](https://projecteuclid.org/euclid.rmi/1218475356)) and I cannot see a claim made in a proof. I don't know if the my question is appropriate to this site, I apologize if not.



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> **Theorem 1.3**. *Let $G$ be a compact $p$-adic analytic group. Then any word $w$ of a free group $F$ has finite width in $G$*.
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**Definition.** We say that $w$ is a $\cal{N}\_p$-word if for every finitely generated pro-$p$
group $H$, $H/\overline{\omega(H)}$ is nilpotent-by-finite, where $\overline{\omega(H)}$ denotes the closure of $\omega(H)$ in $H$.



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> **Theorem 3.1**. *Let $\omega$ be a $\cal{N}\_p$-word and $G$ a finitely generated pro-$p$ group.
> Then $\omega(G)$ is closed.*
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Set $d = d(G)$ and $H$ a free pro-$p$ group on generators $x\_1,...,x\_d,z$. We have that $\gamma\_n(H^{p^t}) \leq \overline{\omega(H)}$ since $\omega$ is a $\cal{N}\_p$-word. The generators $y\_1,...,y\_s$ of $\overline{\langle x\_1,...,x\_d \rangle^{p^t}}$ are pro-$p$ words in $x\_i$. So he applies the theorem 1.3 to get $k$ such that, for every $i\_1,...,i\_n \in \{1,...,s\}$,
$$[z,y\_{i\_1},...,y\_{i\_n}] \equiv v\_{i\_1,...,i\_n} \pmod{\gamma\_{n+2}(H^{p^t})}$$
where $v\_{i\_1,...,i\_n}$ is a product of at most $k$ $\omega$-values in $H$.


I suppose that he works on the quotient group, but I cannot see the conditions to apply the theorem 1.3. Whence the question:


*Why is every nilpotent-by-finite finitely generated pro-p-group always $p$-adic analytic?*



Why is every nilpotent-by-finite finitely generated pro-p-group always $p$-adic analytic
This is indeed true: every finitely generated nilpotent-by-finite (= virtually nilpotent) pro-$p$-group is $p$-adic analytic.


Since every finitely generated nilpotent group has all its subgroup finitely generated, a finitely generated nilpotent profinite group $G$ has a composition series by closed subgroups in which each successive quotient is procyclic. If $G$ is moreover pro-$p$, it follows that each successive quotient is either isomorphic to $\mathbf{Z}\_p$, or a quotient thereof (which is finite). So these successive quotients are $p$-adic analytic. Since being $p$-adic analytic is stable under taking extensions, it follows that every finitely generated nilpotent pro-$p$-group is $p$-adic analytic, and the same follows with "nilpotent-by-finite".


