Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Georg Lehner

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 $K_{-1}( \mathbb{Z}[G] )$.

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

Abstract

We outline how the group for a finite group can be computed using the computer language and compile a table of all groups of order less than that have torsion in .

Paper Structure

This paper contains 9 sections, 3 theorems, 9 equations, 9 tables.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Computing $K_{-1} \mathbb{Z} [G]$ for finite groups using GAP
  4. The free rank $r(G)$
  5. The $\mathbb{Z}/2$ rank $s(G)$
  6. Wrapping things up
  7. $K_{-1} \mathbb{Z} [G]$ of all groups of order $\leq 28$
  8. All groups of order $\leq 100$ with non-trivial torsion in $K_{-1} \mathbb{Z} [G]$
  9. Methodology

Key Result

Theorem 1.1

Let $G$ be finite. The group $K_{-1} \mathbb{Z} G$ has the form where and $s$ is equal to the number of irreducible $\mathbb{Q}$-representations $I$ with even Schur index $m(I)$ but odd local Schur index $m_p(I)$ at every prime $p$ dividing the order of $G$.

Theorems & Definitions (6)

  • Theorem 1.1: Carter, carternegktheory
  • Remark 1.2
  • Theorem 2.1: Berman, berman, see also reiner
  • Lemma 2.2: Magurn magurn_negative, see Lemma 1.
  • Remark 2.3
  • Remark 2.4