Computing the negative $K$-theory of finite groups of order $\leq 100$

TL;DR

This paper demonstrates how to compute the negative K-theory of finite groups of order up to 100 using GAP, providing a comprehensive table of groups with torsion in their K_{-1} groups.

Contribution

It introduces a computational method for determining the negative K-theory of finite groups and compiles a detailed classification for groups of order less than 100.

Findings

Identified all finite groups of order less than 100 with torsion in K_{-1}(Z[G)]

Developed a GAP-based computational approach for negative K-theory

Produced a comprehensive table of relevant groups and their K-theory properties

Abstract

We outline how the group $K_{-1}( \mathbb{Z}[G] )$ for a finite group $G$ can be computed using the computer language $GAP$ and compile a table of all groups $G$ of order less than $100$ that have torsion in $K_{-1}( \mathbb{Z}[G] )$.