- Lectures: MWF 12:05pm – 12:55pm Carver 124
- Office hours: MF 3:10pm – 5:00pm Carver 470

- Main text: Dummit and Foote, Chapters 0–9.
- Notes on Abstract Algebra
- Periodic Table of Finite Simple Groups

Each Friday a homework set is due, except first week, last week and the week of an exam. There will be two exams and one final.

Dates for the exams are:

Exam 1: Friday Sep 30 Review Problems for Exam 1

Exam 2: Wednesday Nov 2 Review Problems for Exam 2 with Some Solutions

Final Exam: Monday Dec 12, noon-2pm in Carver 124 Review Problems for Final Exam with Some Solutions

- (due 9/2)
**§0.1:**4, 5, 7;**§0.2:**2, 10, 11;**§0.3:**15(a)(b);**§1.1:**7, 12, 25, 31 - (due 9/9)
**§1.2:**1(a)(b), 3, 4, 5, 17, 18;**§1.3:**4, 8;**§1.4:**7, 10, 11 - (due 9/16) Prove \((\mathbb{H}_{\mathbb{Z}})^\times=\{\pm 1,\pm i,\pm j,\pm k\}\);
**§1.3:**2, 3, 11, 14;**§1.4:**1, 2;**§1.6:**2, 4, 7, 10, 11, 20 - (due 9/23)
**§1.7:**8;**§2.1:**1(d), 3, 8;**§2.2:**6, 8;**§2.3:**1, 3, 11;**§2.4:**2, 3, 14;**§3.5:**12 - (due 10/7)
**§3.1:**1, 12, 29, 41;**§3.2:**4, 14, 15, 22;**§3.3:**9 - (due 10/14)
**§4.1:**1, 9;**§4.2:**1(a), 4;**§4.3:**11, 19, 20, 21, 30 - (due 10/21)
**§4.4:**1, 3, 13;**§4.5:**7, 13, 17, 22, 32 - (due 10/28) Let \(G=\mathbb{Z}^3/\big(\mathbb{Z}(24,18,21)+\mathbb{Z}(27,-15,12)\big)\). Find a direct product of cyclic groups isomorphic to \(G\).
**§5.2:**2(a)(b)(c), 3(a)(b)(c), 13;**§5.4:**10;**§5.5:**7 - (due 11/11)
**§7.1:**7, 23;**§7.2:**3(b)(c), 9(c), 12 - (due 11/18)
**§7.3:**2, 22, 26, 29, 35;**§7.4:**7, 11, 30, 33 - (due 12/2)
**§7.5:**3, 5;**§7.6:**1, 3, 6, 10;**§8.2:**4, 6, 8

Chapters 0–3

- Equivalence relations and partitions. Integers. Congruence.
- Modular arithmetic. Definition of monoids and groups. Uniqueness of inverse. Examples.
- The group of invertible elements in a monoid. Examples. Generalized Associativity Law. Conventions and notation.
- Cancellation rules, order of an element. Dihedral group.
- Euclidean space, orthogonal group. Linear representation of the dihedral group.
- The group of permutations of a set. Elements of the dihedral group as permutations of the set of vertices. Two-line notation and composition of permutations.
- Symmetric group: Cycles, length, transpositions. Cycle decomposition and its applications. Conjugating cycles.
- Rings, commutative rings, division rings, fields. The ring of quaternions over the real numbers and over the integers. The quaternion group. Homomorphisms. Isomorphic groups.
- Group actions and permutation representations. Left and right regular action, conjugation action. Subgroups, subgroup criterion.
- Centralizer, center, and normalizer subgroups. Kernel of a homomorphism. Determinant and permutation matrices. Alternating group. Stabilizer subgroup. Kernel of an action.
- The subgroup generated by a subset of a group. Cyclic groups. Order of powers of elements.
- Cyclic groups: isomorphisms and subgroups. Lattice of subgroups.
- Cosets. Quotient groups.
- Lagrange's Theorem. Kernels of homomorphisms vs normal subgroups.
- Isomorphism Theorems.
- (Exam 1 Review)

Chapters 4–5, 6.3

- G-sets and the Orbit-Stabilizer Theorem. Cayley's Theorem.
- G acting on itself by conjugation. Class Equation. Center of a p-group is nontrivial.
- Integer partitions and conjugacy classes of the symmetric group.
- Automorphism groups, inner automorphisms, normalizer-mod-centralizer embeds in automorphism group of a subgroup.
- Sylow's Theorem.
- Sylow's Theorem (continued).
- Applications of Sylow's Theorem.
- Fundamental Theorem of Finitely Generated Abelian Groups.
- Semidirect Products.
- Application to the construction of non-abelian groups.
- Free groups and groups given by a presentation.
- (Exam 2 Review)

Chapters 7–9

- Definition of rings, examples, subrings.
- Basic properties, integral domains and fields, monoid rings.
- Subring generation, algebras, the Weyl algebra. Ring homomorphisms and ideals.
- Quotient rings and the isomorphism theorems for rings.
- Prime ideals and integral domains; maximal ideals and fields.
- Universal property of group rings, augmentation ideal. Rings of fractions and their universal property.
- The Remainder Theorem.
- EDs are PIDs. Nonzero prime ideals in a PID are maximal.
- PIDs are UFDs.
- Irreducible elements in polynomial rings and quadratic integers.
- Gauss' Lemma. R[x] is a UFD iff R is a UFD.
- Eisenstein's Irreducibility Criterion.
- (Final Exam Review)

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