Erika Berger: From Abstraction to Action: Exploring Symmetry through Group and Representation Theory

Abstract

This thesis examines the mathematical structure of symmetry found throughout group theory and representation theory. Using group theory as a foundation, it formalizes groups as abstract models of symmetry to develop key algebraic ideas. Representation theory is then introduced as a means of concretely realizing these symmetries through linear transformations on vector spaces. The final chapter synthesizes these ideas by defining symmetry as invariance under transformation and illustrating how representation theory gives form to the symmetries encoded in group theory. Throughout, the study emphasizes symmetry not only as a mathematical concept but as a unifying principle across algebraic structures.