This is a page for Math 411 Intro to Abstract Algebra I in Fall 2026 at UMass Amherst.
Instructor. Peize Liu.
Office: LGRT 1242
Email: peizeliu@umass.edu
Grader. Riley Burton.
Lecture schedule. 9 September – 14 December, 2026.
Section 1: Monday/Wednesday/Friday, 11:15 – 12:05, LGRT Room 141;
Section 2: Monday/Wednesday/Friday, 13:25 – 14:15, LGRT Room 141.
Office hours. Wednesday 4:00 – 5:30 pm and Thursday 3:00 – 4:30 pm, LGRT 1242.
Prerequisites. Math 235 and either Math 300 or CS 250.
Textbook. Abstract Algebra with Applications by Audrey Terras.
Examinations. There will be two mid-term exams and a final exam. The mid-term exams will be in-class (14 October, Wednesday and 18 November, Wednesday) and the final exams are scheduled as follows:
Section 1: 17 December 2026, Thursday, 10:30 – 12:30, in LGRT Room 141;
Section 2: 22 December 2026, Tuesday, 15:30 – 17:30, in LGRT Room 141.
Grading policy. Grades for the course will be determined by:
Homework: 25%; two lowest homework marks will be dropped.
Mid-term 1: 20%.
Mid-term 2: 20%.
Final exam: 35%.
Recap on basic set theory and arithmetic.
Group axioms. Abelian groups.
Examples of groups: transformation groups, dihedral groups, cyclic groups, and matrix groups.
Subgroups and generators. Orders of groups and elements. Isomorphisms.
Structure of cyclic groups. Modular arithmetic. gcd and lcm. Bézout’s lemma.
Product of groups. Chinese remainder theorem.
Permutations and symmetry groups.
Equivalence relations. Cosets. Lagrange's theorem. Fermat's little theorem.
Homomorphisms. Kernels and images.
Normal subgroups. Quotient groups. Isomorphism theorems.
Classification of finitely generated Abelian groups (without proof).
Group actions on sets. Orbits and stabilizers. Orbit–stabilizer theorem. Conjugation.
Orbit counting lemma and applications in combinatorics.
Cayley's theorem. Symmetry groups of some Platonic solids.
Sylow theorems (without proof) and their applications.
Applications in public-key cryptography.
Syllabus
Lecture Notes
Mid-Term 1 Revision Guide
Mid-Term 2 Revision Guide
Sample Questions for Final Exam
Problem Sheet 1: Preliminaries, group axioms
Problem Sheet 2: Examples of groups
Problem Sheet 3: Subgroups and generators
Problem Sheet 4: Cyclic groups and permutation groups
Problem Sheet 5
Problem Sheet 6
Problem Sheet 7
Problem Sheet 8
Problem Sheet 9
Problem Sheet 10