Powerfully embedded subgroups of extensions of powerful pro-$p$ groups
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917303960993792 |
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| author | Kalithasan, Sathasivam Mavely, Tony N. Thomas, Viji Z. |
| author_facet | Kalithasan, Sathasivam Mavely, Tony N. Thomas, Viji Z. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_15240 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Powerfully embedded subgroups of extensions of powerful pro-$p$ groups Kalithasan, Sathasivam Mavely, Tony N. Thomas, Viji Z. Group Theory 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. |
| title | Powerfully embedded subgroups of extensions of powerful pro-$p$ groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2503.15240 |