# Fall 2022

### Time and Place

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

### Texts and Resources

• Main text: Dummit and Foote, Chapters 0–9.

### Examination

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

### Homework Sets

1. (due 9/2) §0.1: 4, 5, 7; §0.2: 2, 10, 11; §0.3: 15(a)(b); §1.1: 7, 12, 25, 31
2. (due 9/9) §1.2: 1(a)(b), 3, 4, 5, 17, 18; §1.3: 4, 8; §1.4: 7, 10, 11
3. (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
4. (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
5. (not due) Review Problems for Exam 1

### Lecture Summary

#### Group Theory Basics

Chapters 0–3

1. Equivalence relations and partitions. Integers. Congruence.
2. Modular arithmetic. Definition of monoids and groups. Uniqueness of inverse. Examples.
3. The group of invertible elements in a monoid. Examples. Generalized Associativity Law. Conventions and notation.
4. Cancellation rules, order of an element. Dihedral group.
5. Euclidean space, orthogonal group. Linear representation of the dihedral group.
6. 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.
7. Symmetric group: Cycles, length, transpositions. Cycle decomposition and its applications. Conjugating cycles.
8. Rings, commutative rings, division rings, fields. The ring of quaternions over the real numbers and over the integers. The quaternion group. Homomorphisms. Isomorphic groups.
9. Group actions and permutation representations. Left and right regular action, conjugation action. Subgroups, subgroup criterion.
10. Centralizer, center, and normalizer subgroups. Kernel of a homomorphism. Determinant and permutation matrices. Alternating group. Stabilizer subgroup. Kernel of an action.
11. The subgroup generated by a subset of a group. Cyclic groups. Order of powers of elements.
12. Cyclic groups: isomorphisms and subgroups. Lattice of subgroups.
13. Cosets. Quotient groups.
14. Lagrange's Theorem. Kernels of homomorphisms vs normal subgroups.
15. Isomorphism Theorems.

Chapters 4–6

Chapters 7–9

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