Powerfully embedded subgroups of extensions of powerful pro-$p$ groups

One of the aims of this paper is to obtain structural results showing that powerful subgroups are abundant in pro-$p$ groups admitting certain powerful quotients. In particular, we obtain an analogue of Baer’s theorem for powerful pro-$p$ groups, namely that the powerfulness of $H/Z_{n-1}(H)$ implies that the $n$th terms of both the lower $p$-series and the lower central series of $H$ are powerfully embedded in $H$. As a consequence, we obtain that if $H$ is a finitely generated pro-$p$ group and $H/Z_n(H)$ is a $p$-adic analytic pro-$p$ group for some positive integer $n$, then $H$ is a $p$-adic analytic pro-$p$ group. We also study crossed squares of powerful $p$-groups, establishing that if $μ\colon M \to G$ is a crossed module with $M$ a finite powerful $p$-group and $G$ a finite $p$-group, and if $μ(M)$ is powerfully embedded in $G$, then both $M \otimes G$ and $M \otimes^{p} G$ are powerful.

💡 Research Summary

The paper investigates the prevalence of powerful subgroups within pro‑p groups that admit certain powerful quotients, and it establishes several structural theorems that extend classical results such as Baer’s theorem to the setting of powerful pro‑p groups.

The first main result, Theorem A (and its variant A′ for p = 2), shows that if the quotient H / Zₙ₋₁(H) is powerful, then the n‑th term of the lower central series γₙ(H) and the n‑th term of the lower p‑series Pₙ(H) are powerfully embedded in H. The proof proceeds by first noting that powerfulness of the quotient forces γ₂(H) ≤ Hᵖ·Zₙ₋₁(H). Using commutator identities and the three‑subgroup lemma, the authors derive the inclusion