Powerful subgroups of extensions of powerful $p$-groups and pro-$p$ groups

Abstract

Let $ H $ be a finitely generated pro-$ p $ group. We prove that, if $H/Z_{n-1}(H)$ is a powerful group, then the $n$-th terms of the lower $p$-series and the lower central series of $H$ are powerfully embedded in $H$. As a consequence, we obtain that if $ 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. Furthermore, we extend these results to pro-$ p $ groups that are not necessarily finitely generated, provided some additional conditions are imposed.

Theorems & Definitions (75)

• Theorem A
• Corollary
• Theorem B
• Theorem C
• Theorem D
• Theorem E
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Theorem 2.4