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Abstract Algebra I — Fall 2026

MATH 3551, Section A · MWF 12:00–12:50, Lafayette L107 · first class Wednesday, September 9
Taylor Dupuy · Innovation E439 · office hours Wednesday 8:30–9:20AM

Lectures

Coverage update (September 28, 2026). Version 2 applies a one-week coverage freeze: material first introduced during the seven calendar days ending on a midterm date moves to the following midterm. It also adds selected Chapter 0 problems to make the foundational material easier to practice; material from §§0.1–0.3 is included on Midterm 1. Sections 1.4 and 1.5 are required independent reading and are also included on Midterm 1. Version 1 remains linked, with the old coverage struck through beside each replacement so that the changes can be compared directly.

Text

• Dummit and Foote — Abstract Algebra, 3rd edition (Wiley), ISBN 0-471-43334-9. Errata.

Exam Schedule

• Midterm 1 (Monday, October 12, in class — §1.1–§3.2 §§0.1–0.3, §§1.1–1.6, §2.1, and §§2.3–2.5)
• Midterm 2 (Friday, November 6, in class — §3.3–§4.4 §1.7, §2.2, and §§3.1–4.3)
• Midterm 3 (Monday, November 30, in class — §4.5–§5.2 §§4.4–5.2)
• Final (Thursday, December 17, 10:30 AM–1:15 PM, Lafayette L107 — cumulative)

Practice Exams

Open an exam below to choose a question sheet or its complete solutions.

Suggested Problems

Each section of the current Version 2 PDF opens with warm-up computations in a specific group — written out in full, so you can start without the book — and then lists the Dummit and Foote exercises below. Do the warm-ups first. Nothing is collected or graded. The Version 1 PDF is preserved for comparison.

Part I — through Midterm 1 (§1.1–§3.2) (§§0.1–0.3, §§1.1–1.6, §2.1, and §§2.3–2.5)

• §0.1 Basics — 4, 6. Added in Version 2 to make the foundational material easier to practice; included on Midterm 1.
• §0.2 Properties of the Integers — 1–5, 10, 11. Added in Version 2 to make the foundational material easier to practice; included on Midterm 1.
• §0.3 Z/nZ: The Integers Modulo n — 3–9. Added in Version 2 to make the foundational material easier to practice; included on Midterm 1.
• §1.1 Basic Axioms and Examples — 1, 5, 11, 20, 22, 25, 26, 32
• §1.2 Dihedral Groups — 1, 2, 3, 4, 5, 6
• §1.3 Symmetric Groups — 1, 2, 3, 4, 5, 14, 15, 18
• §1.4 Matrix Groups — 1, 2, 3, 4, 5, 7. Required independent reading; included on Midterm 1.
• §1.5 The Quaternion Group — 1, 2, 3. Required independent reading; included on Midterm 1.
• §1.6 Homomorphisms and Isomorphisms — 1, 2, 4, 6, 7, 9, 14, 20
• §1.7 Group Actions — 1, 2, 3, 4, 5, 6, 12 Moved to Part II; this material will not be on Midterm 1.
• §2.1 Subgroups: Definition and Examples — 1, 2, 3, 4, 5, 6, 8, 10
• §2.2 Centralizers and Normalizers, Stabilizers and Kernels — 1, 2, 3, 4, 5, 6, 7, 10 Moved to Part II; this material will not be on Midterm 1.
• §2.3 Cyclic Groups and Cyclic Subgroups — 1, 2, 3, 4, 5, 8, 10, 16
• §2.4 Subgroups Generated by Subsets of a Group — 1, 2, 3, 6
• §2.5 The Lattice of Subgroups of a Group — 1, 2, 3, 4, 5, 6
• §3.1 Quotient Groups and Homomorphisms: Definitions and Examples — 1, 2, 3, 4, 6, 7, 14, 36, 37 Moved to Part II; this material will not be on Midterm 1.
• §3.2 More on Cosets and Lagrange's Theorem — 1, 2, 4, 5, 6, 8, 11, 16 Moved to Part II; this material will not be on Midterm 1.

Part II — through Midterm 2 (§3.3–§4.4) (§1.7, §2.2, and §§3.1–4.3)

• §1.7 Group Actions — 1, 2, 3, 4, 5, 6, 12
• §2.2 Centralizers and Normalizers, Stabilizers and Kernels — 1, 2, 3, 4, 5, 6, 7, 10
• §3.1 Quotient Groups and Homomorphisms: Definitions and Examples — 1, 2, 3, 4, 6, 7, 14, 36, 37
• §3.2 More on Cosets and Lagrange's Theorem — 1, 2, 4, 5, 6, 8, 11, 16
• §3.3 The Isomorphism Theorems — 1, 2, 3, 7
• §3.4 Composition Series and the Hölder Program — 1, 2, 3, 4, 5
• §3.5 Transpositions and the Alternating Group — 1, 2, 3, 4, 5, 6, 7
• §4.1 Group Actions and Permutation Representations — 1, 2, 3, 4, 7, 8, 10
• §4.2 Cayley's Theorem — 1, 2, 3, 4, 7, 8, 9, 10
• §4.3 The Class Equation — 1, 2, 3, 4, 5, 6, 13
• §4.4 Automorphisms — 1, 2, 3, 4 Moved to Part III; this material will not be on Midterm 2.

Part III — through Midterm 3 (§4.5–§5.2) (§§4.4–5.2)

• §4.4 Automorphisms — 1, 2, 3, 4
• §4.5 The Sylow Theorems — 1, 2, 3, 4, 5, 13, 19, 24, 26
• §4.6 The Simplicity of A_n — 1, 2
• §5.1 Direct Products — 1, 2, 3, 4
• §5.2 The Fundamental Theorem of Finitely Generated Abelian Groups — 1, 2, 3

Part IV — final only (§5.4, §5.5, §6.1)

• §5.4 Recognizing Direct Products — 1, 2, 3, 4
• §5.5 Semidirect Products — 1, 2
• §6.1 p-groups, Nilpotent Groups, and Solvable Groups — 1, 2

Previous versions of this course

• Abstract Algebra I — UVM — Fall 2020
• Abstract Algebra I — UVM — Fall 2019

Notes

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