A geometric invariant of a finite group

TL;DR

This paper investigates a geometric invariant associated with finite groups in the context of algebraic stacks, demonstrating triviality in many cases including symmetric groups, and providing counterexamples where it is non-trivial.

Contribution

It introduces and studies a new geometric invariant of finite groups within the Grothendieck group of algebraic stacks, including proofs of triviality and examples of non-trivial cases.

Findings

The class is trivial for all symmetric groups.

Counterexamples show the class can be non-trivial.

The study connects to Noether's problem through specific examples.

Abstract

We study the class of the classifying stack of a finite group in a Grothendieck group of algebraic stacks introduced previously. We show that this class is trivial in a number of examples most notably for all symmetric groups. We also give some examples where it is not trivial. The latter uses counterexamples of Saltman and Swan to the problem of Noether.