Finite groups of low essential dimension

Abstract

Informally, essential dimension is the minimal number of parameters required to define an algebraic object. This invariant has numerous connections to Galois cohomology, linear algebraic groups and birational geometry. In particular, the essential dimension of finite groups has connections to the Noether problem, inverse Galois theory and the simplification of polynomials via Tschirnhaus transformations. This thesis studies finite groups of low essential dimension using methods from birational geometry. Specifically, the main results are a classification of finite groups of essential dimension 2, and a proof that the alternating and symmetric groups on 7 letters have essential dimension 4.