# Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

## MAT 150A: Modern Algebra

Approved: 2003-04-01 (revised 2012-08-01, Vazirani, Kuperberg, and Schwarz)
Suggested Textbook: (actual textbook varies by instructor; check your instructor)
Algebra by Michael Artin; Addison Wesley; 2nd edition (August 13, 2010); ISBN-10: 0132413779; Price ranges from \$111.00 to \$145.00.
Search by ISBN on Amazon: 978-0132413770
Suggested Schedule:
 Week 1 Permutations and permutation matrices; Determinants (optional, covered in MAT 67); the definition of a group [option – Finite fields from Chap. 3 and GLn(Fp), SLn(Fp) as examples of groups]. Week 2 Subgroups; Homomorphisms; Isomorphisms (might review diagonalization). Week 3 Focus on examples (Dn, Sn, An, in particular for n = 3,4, the groups of order 8, cycle notation and conjugation in Sn); Cyclic groups, introduction to fields, rings and groups; Cosets. Week 4 Products of groups; Quotient groups; Modular arithmetic. Week 5 Orthogonal matrices and rotation (systems of differential equations and matrix exponentials; Optional, covered in MAT 67); Symmetry of plane figures; The group of motions of the plane. Week 6 Finite groups of motions; Discrete groups of motions/the wallpaper patterns. Week 7 Group operations (focus on examples); Operation of cosets; The Counting Formula (focus on examples); Application of Burnside’s Formula (optional; Papantonopoulou is a good reference). Week 8 Finite Subgroups of the rotation group (Tetrahedral group); Operations of a group on itself; Class equation. Week 9 Operations on subsets; Sylow Theorems; The Groups of order 12. Week 10 Computations in the Symmetric Group; The free group; Generators and relations.
If there is time, one may cover select parts of Chapter 9.
Learning Goals:

In this course students will:

1. Understand the major theorems and their proofs;
2. Prove theorems in general algebraic settings;
3. Apply general algebraic theorems in specific instances; and
4. Understand the fundamental principles underlying the algebra with which they are familiar.

In particular, by the end of this course, students will develop intuition about the objects of modern algebra, have more finely developed proof-writing skills, and skills that enable them to better read, understand, and communicate mathematics.

They will have learned to interpret and analyze fundamental principles and theory concerning basic algebraic structures, including groups, product structures, generators and relations, homomorphisms, and isomorphisms.

They will be able to construct accurate, meaningful examples of algebraic structures, and relate theory from this course to previous mathematical knowledge.

Students should understand how to analyze and algebraically describe the symmetry of plane figures and how the symmetries of different figures are related.

They will be able to prove and analyze the truth of algebraic statements, and to clearly communicate mathematical ideas, in both written and oral form.