- 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.

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.

Estimated dates for the exams are:

Exam 1: Friday, Sep 30

Exam 2: Friday, Oct 28

- (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 - (not due) Review Problems for Exam 1

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.

Chapters 4–6

Chapters 7–9

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