ABSTRACT
Though this textbook waited until this point to introduce group actions, from a historical perspective, group actions came first and motivated group theory. Mathematicians did not define groups ex nihilo and study their properties from their axioms. Evariste Galois, often credited with defining a group in the modern sense (Definition 3.2.1), formalized the axioms of groups while studying properties of symmetry among roots of a polynomial. Subsequent work by mathematicians simultaneously revealed the richness of the algebraic structure of groups, discovered group-theoretic patterns in many areas, and developed Galois theory, which applies group actions to the study of polynomial equations. This textbook covers group actions in this chapter and Galois theory in Chapter 11.
