Actions of Groups: A Second Course in Algebra

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.

這本現代代數的第二門課程以群作用的統一概念為基礎，介紹了更深層的代數工具，這些工具在第一門課中只是稍作提及，例如有限簡單群的成功分類以及群在多項式方程解的過程中所扮演的角色。由於群可以作為集合的置換、向量空間的線性變換或者是一個域的自同構，因此從這些觀點出發，群的更深層結構可能會浮現出來，可以區分兩個不同的群，或者可以證明一個多項式方程可以用根式解。通過發展這些群作用的性質，讀者將接觸到像Sylow定理及其應用、Galois理論和表示論等重要的代數主題。回顧和擴展第一門課的熱身章節以及積極學習模塊有助於學生更深入地理解這些概念。