Two inquiries about finite groups and well-behaved quotients

TL;DR

This thesis explores the singularities of quotient spaces under finite group actions and the Cohen-Macaulay property of invariant rings, providing classifications for supersolvable groups and specific permutation groups.

Contribution

It introduces a class of finite groups with well-behaved quotient singularities, proves supersolvable groups belong to this class, and characterizes when invariant rings are Cohen-Macaulay for certain permutation groups.

Findings

Supersolvable groups have quotients with abelian quotient singularities.

Invariant rings are Cohen-Macaulay for groups generated by transpositions, double transpositions, and 3-cycles.

Nonabelian finite simple groups do not belong to the class with well-behaved quotient singularities.

Abstract

This thesis addresses questions in representation and invariant theory of finite groups. The first concerns singularities of quotient spaces under actions of finite groups. We introduce a class of finite groups such that the quotients have at worst abelian quotient singularities. We prove that supersolvable groups belong to this class and show that nonabelian finite simple groups do not belong to it. The second question concerns the Cohen-Macaulayness of the invariant ring $\mathbb{Z}[x_1,\dots,x_n]^G$, where $G$ is a permutation group. We prove that this ring is Cohen-Macaulay if $G$ is generated by transpositions, double transpositions, and 3-cycles, and conjecture that the converse is true as well.