The minimal degree of permutation representations of finite groups

TL;DR

This thesis investigates the minimal degree of permutation representations of finite groups, providing explicit formulas for abelian groups, analyzing behavior under direct products, and introducing a compression ratio to measure embedding efficiency.

Contribution

It offers an explicit formula for abelian groups' minimal permutation degree, studies how this property behaves under direct products, and introduces the compression ratio as a new measure of embedding optimality.

Findings

Explicit formula for abelian groups' permutation degree

Behavior of permutation degree under direct products

Introduction of the compression ratio as a new metric

Abstract

In this thesis we study the following property of a finite group G: the minimal number n such that G embeds in Sn. We start with an explicit formula for the number n for abelian groups. Then, we study the behavior of this group property in respect to direct products. Finally, we define and explore the "compression ratio" of a finite group G which measures how much better the best embedding is relative to the embedding given by Cayley's theorem.