Actions of Groups

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.

• Presents unified narratives around fundamental concepts in modern algebra, allowing readers to anticipate and develop methods that are related but widely applicable
• Features active learning modules, “Getting to Know…,” where readers work out ideas for themselves, offering opportunities for mathematical discourse among students in group settings
• Discusses representation theory – an often-overlooked topic for undergraduates – while illustrating its importance, depth, and remarkable applications
• Provides a more motivated narrative of Galois theory, developed with group actions to hand, and exposes readers to concrete examples of Galois theory in action