# Cohomology of uniserial p-adic space groups with cyclic point group

## TL;DR

This paper investigates the mod p cohomology of quotients of uniserial p-adic space groups with cyclic point groups, showing that large enough quotients share isomorphic cohomology groups, revealing structural similarities.

## Contribution

It proves that all sufficiently large quotients of certain uniserial p-adic space groups have isomorphic mod p cohomology groups, highlighting a uniform cohomological behavior.

## Key findings

- Large quotients have isomorphic mod p cohomology groups
- Cohomology groups of maximal class pro-p groups are isomorphic as _p-modules
- Structural cohomology properties are preserved in large quotients

## Abstract

Let $p$ be a fixed prime number and let $R$ denote a uniserial $p$-adic space group of dimension $d_x=(p-1)p^{x-1}$ and with cyclic point group of order $p^x$. In this short note we prove that all the quotients of $R$ of size bigger than or equal to $p^{d_x+x}$ have isomorphic mod $p$ cohomology groups. In particular, we show that the cohomology groups of sufficiently large quotients of the unique maximal class pro-$p$ group are isomorphic as $\F_p$-modules.