### Textbook and notes

Abstract Algebra: Theory and Applications, by Thomas W. Judson.
Notes on Modern Algebra I by Patrick Gallagher.

### Syllabus in brief

Sets and maps. Divisors and basic arithmetic. Groups and Monoids. Subgroups and cosets, Lagrange's theorem. Normal subgroups, factor groups, isomorphism theorems. Permutations, symmetric and alternating groups. Actions of groups on sets. Conjugacy classes, automorphisms. Sylow's theorems and their applications. Groups and geometry. Presenting a group via generators and relations.

### Detailed syllabus:

Lectures 1-2 (Wed Jan 18, Mon Jan 23) Set theory. Judson Chapter 1 (Preliminaries) and Gallagher Chapters 4-6 (Algebra of sets, Algebra of maps, Inverse maps, partitions). Sets, operations on sets (union, intersection, complement). Counting subsets (binomial coefficients). Cartesian product and the power set. De Morgan's Laws. Maps of sets: surjections, injections, bijections. Associativity of composition. Identity map. Partitions of sets and equivalence relations.

Lecture 3 (Wed Jan 25) Modular arithmetic and residues. Read Judson Chapter 2 (The Integers) and Gallagher Chapters 1-2 (Divisors, Unique factorization). Some of the background material for sets and arithmetic, as well as introduction to proofs, can be found in MathDoctorBob's video course Math Major Basics.

Lecture 4 (Mon Jan 30) Binary operations, associativity. Notion and basic properties of a group. Multiplication table. Read Judson Chapter 3 (Groups). Gallagher (Monoids, Groups) emphasizes monoids. You'll find more examples, including multiplication tables for groups, in Howie in Chapter 1 and Sections 2.1, 2.2.

Lecture 5 (Wed Feb 1) Subgroups, examples of subgroups (Judson Section 3.3). Quiz 1.

Lecture 6 (Mon Feb 6) Properties of groups (Judson 3.2, starting Proposition 3.17). Powers of an element (Theorem 3.23). Cyclic groups (Judson Section 4.1 up to and including Theorem 4.9).

Lecture 7 (Wed Feb 8) Subgroups of cyclic groups (Judson Section 4.1 starting Theorem 4.10). Multiplicative group of complex numbers and its subgroups (Judson Section 4.2). Direct product of groups. The group of invertible residues modulo n under multiplication.

Lecture 8 (Mon Feb 13) Symmetric group, permutations and cycles (Judson Section 5.1 up to and including Remark 5.11). Multiplication of permutations in cycle notation. Left and right cosets (Judson 6.1).

Mon Feb 15 Midterm 1.

Lecture 9 (Mon Feb 20) Left and right cosets (Judson 6.1). Lagrange's theorem (Section 6.2, up to but not including Remark 6.14). Section 6.3 (Fermat's and Euler's theorems).

Lecture 10 (Wed Feb 22) Judson Section 5.1 continued: Transpositions, odd and even permutations, alternating group. Some applications of Lagrange's theorem. Quiz 2.

Lecture 11 (Mon Feb 27)Section 6.2 starting with Remark 6.14: Subgroups of alternating group A4. We discussed conjugacy classes in groups (in Judson this is considered only within a more general setup of Section 14) and showed that two permutation are conjugate in the symmetric group iff they have the same cycle type. We then discussed isomorphisms of groups (Judson Section 9.1, skipping Cayley's theorem that any finite group is a subgroup of the symmetric group), followed by External Direct Products in Section 9.2, with applications to decomposing cyclic groups as direct products.

Lecture 12 (Wed Mar 1) Internal direct products (2nd part of Section 9.2). Section 10.1: Normal subgroups and factor groups. Section 11.1: Group homomorphisms.

Lecture 13 (Mon Mar 6) Isomorphism theorems (Section 11.2). I also recommend reading alternative exposition, with diagrams of subgroups, in Gallagher Sections 11 and 12. Quaternion group (Example 3.15 in Section 3.2).

Lecture 14 (Wed Mar 8) Isomorphism theorems continued. Dihedral groups (Section 5.2). Quiz 3.

SPRING BREAK!

Lecture 15 (Mon Mar 20) Quaternion group. Characters of finite abelian groups (Gallagher Section 14). Centers of groups.

Lecture 16 (Wed Mar 22) Characters continued (Gallaher Sections 14, 15).

Lecture 17 (Mon Mar 27) Classification of finite abelian groups (Gallagher Section 15, Judson Section 13.1).

Midterm 2 (Wed Mar 29)

Lecture 18 (Mon Apr 3) Group actions (Judson 14.1, Gallagher Section 16).

Lecture 19 (Wed Apr 5) The class equation, automorphisms (Judson 14.2, Gallagher Section 17). Groups of prime power order (Gallagher Section 18, Judson Section 15.1).

Lecture 20 (Mon Apr 10) Sylow theorems (Judson Section 15.1, Gallagher Section 18).

Lecture 21 (Wed Apr 12) Sylow theorems, applications (Judson Section 15.2, Gallagher Section 19). Quiz 4.

Lecture 22 (Mon Apr 17) Solvable groups (Gallagher Section 12 starting page 3), commutator subgroup, solvability of S4, simple groups. Simplicity of alternating group A5.

Lecture 23 (Wed Apr 19) Small groups are not simple (Gallagher Section 19). Burnside-Cauchy-Frobenius theorem (Gallagher, end of Section 16) and Judson Section 14.3, applications to orbit counting.

Lecture 24 (Mon Apr 24) Reflections and rotations. Symmetry groups of 3-dimensional solids (Gallagher Sections 21-23). Quiz 5.

Lecture 25 (Wed Apr 26) Symmetry groups of solids continued (Gallagher Sections 21-23). More orbit counting (Section 24).

Lecture 26 (Mon May 1) Review.

FINAL EXAM (Mon May 8) 1:10-4pm in Math 203.