The powerful class of groups

TL;DR

This paper investigates pro-$p$ groups with finite powerful class, demonstrating their $p$-adic analyticity and characterizing their structure for small classes, while also establishing finiteness results for finite $p$-groups with fixed coclass and powerful class.

Contribution

The paper proves that pro-$p$ groups of finite powerful class are $p$-adic analytic and characterizes their structure in cases of small class, also showing finiteness of certain finite $p$-groups.

Findings

Pro-$p$ groups of finite powerful class are $p$-adic analytic.

Structural descriptions for groups with small powerful class.

Finiteness of finite $p$-groups with fixed coclass and powerful class.

Abstract

Pro-$p$ groups of finite powerful class are studied. We prove that these are $p$-adic analytic, and further describe their structure when their powerful class is small. It is also shown that there are only finitely many finite $p$-groups of fixed coclass and powerful class.