Course Description:
In this upper level study, explore the theory and applications of the algebraic structures known as groups. Topics covered in this course include: an introduction to groups; the dihedral groups; homomorphisms and isomorphisms; subgroups and cyclic subgroups; group actions; permutations; cosets and Lagrange's Theorem, Cayley's Theorem; the Sylow Theorems and the Fundamental Theorem of Finitely Generated Abelian Groups. Following this thorough investigation of group theory, students will begin to explore the basic ideas of ring theory.
The primary audience for this course is students who wish to concentrate in either mathematics or applied mathematics. Students interested in various fields which have a strong connection to this branch of mathematics (such as music theory, physics, chemistry, computer science, or the cognitive sciences) may also be interested in this course.
Prior to enrolling in this course, students should be fluent in the foundations of mathematics and mathematical proof: logic, methods of proof (both inductive and deductive), sets, relations and functions. This knowledge may be obtained from a course such as Proof and Logic or Discrete Mathematics, for example. Students should also be familiar with matrices and determinants; this knowledge can be obtained from a course such as Linear Algebra.
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