Computing a spectral sequence of finite Heisenberg groups of prime power order

TL;DR

This paper analyzes the Lyndon-Hochschild-Serre spectral sequence for finite Heisenberg groups of prime power order, showing it collapses early and exhibits stability across different group sizes.

Contribution

It provides the first detailed computation of the spectral sequence for these groups, revealing collapse at the third page and isomorphism stability from the second page onward.

Findings

Spectral sequence collapses at the third page.

Spectral sequences are isomorphic from the second page for fixed p.

Results apply to Heisenberg groups modulo p^n for primes p ≥ 5.

Abstract

Let $p\geq 5$ be a prime number, let $n\geq 2$ be a natural number and let $\text{Heis}(p^n)$ denote the Heisenberg group modulo $p^n$. We study the Lyndon-Hochschild-Serre spectral sequence $E(\text{Heis}(p^n))$ associated to $\text{Heis}(p^n)$ considered as a split extension, and show that, $E(\text{Heis}(p^n))$ collapses in the third page. Moreover, for a fixed $p$, the spectral sequences $E(\text{Heis}(p^n))$ are isomorphic from the second page on.