Actions of Groups

An undergraduate text with an active learning approach introducing representation theory and Galois theory topics using group actions.

This undergraduate text goes beyond the basics of modern algebra by presenting the fundamental topics of representation theory and Galois theory in a manner guided by the notion of actions of groups. Active learning modules and chapters that extend the first course support the transition to deeper ideas.Using the unifying notion of group actions, this second course in modern algebra introduces the deeper algebraic tools needed to get into topics only hinted at in a first course, like the successful classification of finite simple groups and how groups play a role in the solutions of polynomial equations. Because groups may act as permutations of a set, as linear transformations on a vector space, or as automorphisms of a field, the deeper structure of a group may emerge from these viewpoints, two different groups can be distinguished, or a polynomial equation can be shown to be solvable by radicals. By developing the properties of these group actions, readers encounter essential algebra topics like the Sylow theorems and their applications, Galois theory, and representation theory. Warmup chapters that review and build on the first course and active learning modules help students transition to a deeper understanding of ideas.