A double coset formula for the genus of a nilpotent group

TL;DR

This paper develops double coset formulas to compute the genus and extended genus of finitely generated nilpotent groups, utilizing automorphisms related to fracture squares.

Contribution

It introduces a novel double coset formula approach for calculating the genus of nilpotent groups based on automorphism concepts.

Findings

Derived explicit double coset formulas for the genus of nilpotent groups

Connected automorphism notions to the computation of group genus

Provided a framework for extended genus calculations

Abstract

We derive double coset formulae for the genus and extended genus of a finitely generated nilpotent group G, using the notions of bounded and bounded above automorphisms of $\prod G_S$, which are defined relative to a fixed fracture square for G.