Upper Bounds on the Torsion Index of Half-Spin Groups

TL;DR

This paper establishes upper bounds for the torsion index of half-spin groups, showing they are generally at most twice that of spin groups, with some exceptions where the bound is higher.

Contribution

It provides the first comprehensive upper bounds for the torsion index of half-spin groups, completing the calculation for all split simple groups.

Findings

Most half-spin groups have torsion indexes at most twice those of spin groups.

Exceptional cases have torsion indexes up to 8 times that of spin groups.

In many cases, the torsion index of half-spin groups equals that of spin groups.

Abstract

The torsion index of split simple groups has been extensively studied, notably by Totaro, who calculated the torsion indexes of the spin groups and $E_{8}$ in [5] and [6], respectively. The aim of this paper is to provide upper bounds for the torsion index of half-spin groups, the only remaining case in the calculation of torsion indexes for split simple groups. We present general upper bounds for the torsion index of half-spin groups, showing that, except for certain exceptional cases, it is at most twice that of the corresponding spin groups. For these exceptional cases, the torsion index is bounded above by at most $2^3$ times that of the spin groups. Our results also reveal that in many cases, the torsion index of half-spin groups coincides with that of the spin groups.